Last night a further generalization of Dr. A. Lakhtakia's
generalized Sierpinski-Pascal to higher combinartorial levels was obtained.
after closely reading the reference paper.
A few of the sequences that are involved were submitted to OEIS.
Pascal level:
%I A154690
%S A154690
2,3,3,5,8,5,9,18,18,9,17,40,48,40,17,33,90,120,120,90,33,65,204,300,
%T A154690
320,300,204,65,129,462,756,840,840,756,462,129,257,1040,1904,2240,2240,
%U A154690
2240,1904,1040,257,513,2322,4752,6048,6048,6048,6048,4752,2322,513
%N A154690 Generalized Sierpinski-Pascal gasket triangular sequence:p =
2; q = 1; t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*Binomial[n, m].
%C A154690 Row sums are:A025192 :
%C A154690 {2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098,...}
%D A154690 A. Lakhtakia,R. Messier, V.K. Varadan,V.V. Varadan, "Use of
combinatorial algebra for diffusion on fractals" ,Physical Review A,
volume34, Number3, Sept 1986,page 2502, (FIG. 3)
%F A154690 p = 2; q = 1; t(n,m)=(p^(n - m)*q^m + p^m*q^(n -
m))*Binomial[n, m].
%e A154690 {2},
%e A154690 {3, 3},
%e A154690 {5, 8, 5},
%e A154690 {9, 18, 18, 9},
%e A154690 {17, 40, 48, 40, 17},
%e A154690 {33, 90, 120, 120, 90, 33},
%e A154690 {65, 204, 300, 320, 300, 204, 65},
%e A154690 {129, 462, 756, 840, 840, 756, 462, 129},
%e A154690 {257, 1040, 1904, 2240, 2240, 2240, 1904, 1040, 257},
%e A154690 {513, 2322, 4752, 6048, 6048, 6048, 6048, 4752, 2322, 513},
%e A154690 {1025, 5140, 11700, 16320, 16800, 16128, 16800, 16320, 11700,
5140, 1025}
%t A154690 Clear[t, p, q, n, m]; p = 2; q = 1;
%t A154690 t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*Binomial[n, m];
%t A154690 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154690 Flatten[%]
%Y A154690 A025192
%K A154690 nonn,tabl
%O A154690 0,1
%A A154690 Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Jan 14 2009
%I A154692
%S A154692
2,5,5,13,24,13,35,90,90,35,97,312,432,312,97,275,1050,1800,1800,1050,
%T A154692
275,793,3492,7020,8640,7020,3492,793,2315,11550,26460,37800,37800,
%U A154692
26460,11550,2315,6817,38064,97776,157248,181440,157248,97776,38064
%N A154692 Generalized Sierpinski-Pascal gasket triangular sequence:p =
2; q = 3; t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*Binomial[n, m].
%C A154692 Row sums are:A020729 :
%C A154692 {2, 10, 50, 250, 1250, 6250, 31250, 156250, 781250, 3906250,
19531250,...}
%D A154692 A. Lakhtakia,R. Messier, V.K. Varadan,V.V. Varadan, "Use of
combinatorial algebra for diffusion on fractals" ,Physical Review A,
volume34, Number3, Sept 1986,page 2502, (FIG. 3)
%F A154692 p = 2; q = 3; t(n,m)=(p^(n - m)*q^m + p^m*q^(n -
m))*Binomial[n, m].
%e A154692 {2},
%e A154692 {5, 5},
%e A154692 {13, 24, 13},
%e A154692 {35, 90, 90, 35},
%e A154692 {97, 312, 432, 312, 97},
%e A154692 {275, 1050, 1800, 1800, 1050, 275},
%e A154692 {793, 3492, 7020, 8640, 7020, 3492, 793},
%e A154692 {2315, 11550, 26460, 37800, 37800, 26460, 11550, 2315},
%e A154692 {6817, 38064, 97776, 157248, 181440, 157248, 97776, 38064,
6817},
%e A154692 {20195, 125010, 356400, 635040, 816480, 816480, 635040,
356400, 125010, 20195},
%e A154692 {60073, 409020, 1284660, 2514240, 3538080, 3919104, 3538080,
2514240, 1284660, 409020, 60073}
%t A154692 Clear[t, p, q, n, m]; p = 2; q = 3;
%t A154692 t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*Binomial[n, m];
%t A154692 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154692 Flatten[%]
%Y A154692 A020729
%K A154692 nonn,tabl
%O A154692 0,1
%A A154692 Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Jan 14 2009
Eulerian numbers level:
%I A154693
%S A154693
2,3,3,5,16,5,9,66,66,9,17,260,528,260,17,33,1026,3624,3624,1026,33,65,
%T A154693
4080,23820,38656,23820,4080,65,129,16302,154548,374856,374856,154548,
%U A154693
16302,129,257,65260,993344,3529360,4998080,3529360,993344,65260,257
%N A154693 Generalized Sierpinski-Pascal-Eulerian gasket triangular
sequence:p = 2; q = 1; t(n,m)=(p^(n - m)*q^m + p^m*q^(n -
m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}].
%C A154693 Row sums are:A000629 :
%C A154693 {2, 6, 26, 150, 1082, 9366, 94586, 1091670, 14174522,
204495126, 3245265146,...}
%D A154693 A. Lakhtakia,R. Messier, V.K. Varadan,V.V. Varadan, "Use of
combinatorial algebra for diffusion on fractals" ,Physical Review A,
volume34, Number3, Sept 1986,page 2502, (FIG. 3)
%F A154693 p = 2; q = 1;
%F A154693 t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*Sum[(-1)^j*Binomial[n
+ 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}].
%e A154693 {2},
%e A154693 {3, 3},
%e A154693 {5, 16, 5},
%e A154693 {9, 66, 66, 9},
%e A154693 {17, 260, 528, 260, 17},
%e A154693 {33, 1026, 3624, 3624, 1026, 33},
%e A154693 {65, 4080, 23820, 38656, 23820, 4080, 65},
%e A154693 {129, 16302, 154548, 374856, 374856, 154548, 16302, 129},
%e A154693 {257, 65260, 993344, 3529360, 4998080, 3529360, 993344,
65260, 257},
%e A154693 {513, 261354, 6314880, 32773824, 62896992, 62896992,
32773824, 6314880, 261354, 513},
%e A154693 {1025, 1046504, 39685620, 299674368, 779049120, 1006351872,
779049120, 299674368, 39685620, 1046504, 1025}
%t A154693 Clear[t, p, q, n, m]; p = 2; q = 1;
%t A154693 t[n_, m_] =(p^(n - m)*q^m + p^m*q^(n -
m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}];
%t A154693 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154693 Flatten[%]
%Y A154693 A000629
%K A154693 nonn,tabl
%O A154693 0,1
%I A154694
%S A154694
2,5,5,13,48,13,35,330,330,35,97,2028,4752,2028,97,275,11970,54360,
%T A154694
54360,11970,275,793,69840,557388,1043712,557388,69840,793,2315,407550,
%U A154694
5409180,16868520,16868520,5409180,407550,2315,6817,2388516,51011136
%N A154694 Generalized Sierpinski-Pascal-Eulerian gasket triangular
sequence:p = 2; q = 3; t(n,m)=(p^(n - m)*q^m + p^m*q^(n -
m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}].
%C A154694 Row sums are:A004123 :
%C A154694 {2, 10, 74, 730, 9002, 133210, 2299754, 45375130, 1007179562,
24840104410,
%C A154694 673895590634,...}
%D A154694 A. Lakhtakia,R. Messier, V.K. Varadan,V.V. Varadan, "Use of
combinatorial algebra for diffusion on fractals" ,Physical Review A,
volume34, Number3, Sept 1986,page 2502, (FIG. 3)
%F A154694 p = 2; q = 3;
%F A154694 t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*Sum[(-1)^j*Binomial[n
+ 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}].
%e A154694 {2},
%e A154694 {5, 5},
%e A154694 {13, 48, 13},
%e A154694 {35, 330, 330, 35},
%e A154694 {97, 2028, 4752, 2028, 97},
%e A154694 {275, 11970, 54360, 54360, 11970, 275},
%e A154694 {793, 69840, 557388, 1043712, 557388, 69840, 793},
%e A154694 {2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550,
2315},
%e A154694 {6817, 2388516, 51011136, 247761072, 404844480, 247761072,
51011136, 2388516, 6817},
%e A154694 {20195, 14070570, 473616000, 3441251520, 8491093920,
8491093920, 3441251520, 473616000, 14070570, 20195},
%e A154694 {60073, 83276472, 4357481076, 46167480576, 164067744672,
244543504896, 164067744672, 46167480576, 4357481076, 83276472, 60073}
%t A154694 Clear[t, p, q, n, m]; p = 2; q = 3;
%t A154694 t[n_, m_] =(p^(n - m)*q^m + p^m*q^(n -
m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}];
%t A154694 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154694 Flatten[%]
%Y A154694 A004123
%K A154694 nonn,tabl
%O A154694 0,1
MacMahon numbers level:
%I A154694
%S A154694
2,5,5,13,48,13,35,330,330,35,97,2028,4752,2028,97,275,11970,54360,
%T A154694
54360,11970,275,793,69840,557388,1043712,557388,69840,793,2315,407550,
%U A154694
5409180,16868520,16868520,5409180,407550,2315,6817,2388516,51011136
%N A154694 Generalized Sierpinski-Pascal-Eulerian gasket triangular
sequence:p = 2; q = 3; t(n,m)=(p^(n - m)*q^m + p^m*q^(n -
m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}].
%C A154694 Row sums are:A004123 :
%C A154694 {2, 10, 74, 730, 9002, 133210, 2299754, 45375130, 1007179562,
24840104410,
%C A154694 673895590634,...}
%D A154694 A. Lakhtakia,R. Messier, V.K. Varadan,V.V. Varadan, "Use of
combinatorial algebra for diffusion on fractals" ,Physical Review A,
volume34, Number3, Sept 1986,page 2502, (FIG. 3)
%F A154694 p = 2; q = 3;
%F A154694 t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*Sum[(-1)^j*Binomial[n
+ 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}].
%e A154694 {2},
%e A154694 {5, 5},
%e A154694 {13, 48, 13},
%e A154694 {35, 330, 330, 35},
%e A154694 {97, 2028, 4752, 2028, 97},
%e A154694 {275, 11970, 54360, 54360, 11970, 275},
%e A154694 {793, 69840, 557388, 1043712, 557388, 69840, 793},
%e A154694 {2315, 407550, 5409180, 16868520, 16868520, 5409180, 407550,
2315},
%e A154694 {6817, 2388516, 51011136, 247761072, 404844480, 247761072,
51011136, 2388516, 6817},
%e A154694 {20195, 14070570, 473616000, 3441251520, 8491093920,
8491093920, 3441251520, 473616000, 14070570, 20195},
%e A154694 {60073, 83276472, 4357481076, 46167480576, 164067744672,
244543504896, 164067744672, 46167480576, 4357481076, 83276472, 60073}
%t A154694 Clear[t, p, q, n, m]; p = 2; q = 3;
%t A154694 t[n_, m_] =(p^(n - m)*q^m + p^m*q^(n -
m))*Sum[(-1)^j*Binomial[n + 2, j]*(m - j + 1)^(n + 1), {j, 0, m + 1}];
%t A154694 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154694 Flatten[%]
%Y A154694 A004123
%K A154694 nonn,tabl
%O A154694 0,1
%I A154696
%S A154696
2,5,5,13,72,13,35,690,690,35,97,5928,16560,5928,97,275,49770,302760,
%T A154696
302760,49770,275,793,420204,4934124,10172736,4934124,420204,793,2315,
%U A154696 3595350,76427820,280500840,280500840,76427820,3595350,2315,6817
%N A154696 Generalized Sierpinski-Pascal-MacMahon gasket triangular
sequence:p = 2; q = 3; p(x,n)=2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2];
t(n,m)=Coefficients(p(x,n)); t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*t(n,m)
%C A154696 Row sums are:
%C A154696 {2, 10, 98, 1450, 28610, 705610, 20882978, 721052650,
28453354370,
%C A154696 1263142915210, 62305874905058,...}
%D A154696 A. Lakhtakia,R. Messier, V.K. Varadan,V.V. Varadan, "Use of
combinatorial algebra for diffusion on fractals" ,Physical Review A,
volume34, Number3, Sept 1986,page 2502, (FIG. 3)
%F A154696 p = 2; q = 3;
%F A154696 p(x,n)=2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2];
%F A154696 t(n,m)=Coefficients(p(x,n));
%F A154696 t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*t(n,m)
%e A154696 {2},
%e A154696 {5, 5},
%e A154696 {13, 72, 13},
%e A154696 {35, 690, 690, 35},
%e A154696 {97, 5928, 16560, 5928, 97},
%e A154696 {275, 49770, 302760, 302760, 49770, 275},
%e A154696 {793, 420204, 4934124, 10172736, 4934124, 420204, 793},
%e A154696 {2315, 3595350, 76427820, 280500840, 280500840, 76427820,
3595350, 2315},
%e A154696 {6817, 31174416, 1157989104, 6978688704, 12117636288,
6978688704, 1157989104, 31174416, 6817},
%e A154696 {20195, 273257970, 17387766000, 164112268320, 449798145120,
449798145120, 164112268320, 17387766000, 273257970, 20195},
%e A154696 {60073, 2414772276, 260247593268, 3735760540608,
15279843455136, 23749342062336, 15279843455136, 3735760540608,
260247593268, 2414772276, 60073}
%t A154696 Clear[t, p, q, n, m, a];
%t A154696 p[x_, n_] = 2^n*(1 - x)^(n + 1)* LerchPhi[x, -n, 1/2];
%t A154696 a = Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]],
x], {n, 0, 10}];
%t A154696 p = 2; q = 3;
%t A154696 t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*a[[n + 1]][[m + 1]];
%t A154696 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154696 Flatten[%]
%K A154696 nonn,tabl
%O A154696 0,1
4th level(1,8,1):
%I A154697
%S A154697
2,3,3,5,24,5,9,138,138,9,17,760,1840,760,17,33,4266,20184,20184,4266,
%T A154697
33,65,24548,210860,376768,210860,24548,65,129,143814,2183652,6233352,
%U A154697 6233352,2183652,143814,129,257,851760,22549616,99411520,149600448
%N A154697 Generalized Sierpinski-Pascal-4th gasket triangular
sequence:p = 2; q = 1; A(n,m)=(3*n - 3*k + 1)A(n - 1, k - 1) + (3*k -
2)A(n - 1, k); t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*A(n+1,m+1)
%C A154697 Row sums are:
%C A154697 {2, 6, 42, 486, 7722, 156006, 3826602, 110419686, 3664093482,
137444066406,
%C A154697 5750512072362,...}
%D A154697 A. Lakhtakia,R. Messier, V.K. Varadan,V.V. Varadan, "Use of
combinatorial algebra for diffusion on fractals" ,Physical Review A,
volume34, Number3, Sept 1986,page 2502, (FIG. 3)
%F A154697 p = 2; q = 1;
%F A154697 A(n,m)=(3*n - 3*k + 1)A(n - 1, k - 1) + (3*k - 2)A(n - 1, k);
%F A154697 t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*A(n+1,m+1)
%e A154697 {2},
%e A154697 {3, 3},
%e A154697 {5, 32, 5},
%e A154697 {9, 234, 234, 9},
%e A154697 {17, 1660, 4368, 1660, 17},
%e A154697 {33, 12186, 65784, 65784, 12186, 33},
%e A154697 {65, 92616, 943500, 1754240, 943500, 92616, 65}, {129,
720390, 13457124, 41032200, 41032200, 13457124, 720390, 129},
%e A154697 {257, 5678660, 192035264, 923107760, 1422449600, 923107760,
192035264, 5678660, 257},
%e A154697 {513, 45086274, 2736272880, 20479237344, 45461436192,
45461436192, 20479237344, 2736272880, 45086274, 513},
%e A154697 {1025, 359306560, 38863293300, 448852224000, 1417337239200,
1939687944192, 1417337239200, 448852224000, 38863293300, 359306560, 1025}
%t A154697 Clear[t, p, q, n, m, A]; A[n_, 1] := 1; A[n_, n_] := 1;
%t A154697 A[n_, k_] := (3*n - 3*k + 1)A[n - 1, k - 1] + (3*k - 2)A[n -
1, k];
%t A154697 p = 2; q = 1;
%t A154697 t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*A[n + 1, m + 1];
%t A154697 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154697 Flatten[%]
%K A154697 nonn,tabl
%O A154697 0,1
%A A154697 Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Jan 14 2009
%I A154698
%S A154698
2,5,5,13,96,13,35,1170,1170,35,97,12948,39312,12948,97,275,142170,
%T A154698 986760,986760,142170,275,793,1585368,22077900,47364480,22077900,
%U A154698
1585368,793,2315,18009750,470999340,1846449000,1846449000,470999340
%N A154698 Generalized Sierpinski-Pascal-4th gasket triangular
sequence:p = 2; q = 3; A(n,m)=(3*n - 3*k + 1)A(n - 1, k - 1) + (3*k -
2)A(n - 1, k); t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*A(n+1,m+1)
%C A154698 Row sums are:
%C A154698 {2, 10, 122, 2410, 65402, 2258410, 94692602, 4670920810,
264961589882,
%C A154698 16990523224810, 1215217470322682,...}
%D A154698 A. Lakhtakia,R. Messier, V.K. Varadan,V.V. Varadan, "Use of
combinatorial algebra for diffusion on fractals" ,Physical Review A,
volume34, Number3, Sept 1986,page 2502, (FIG. 3)
%F A154698 p = 2; q = 3;
%F A154698 A(n,m)=(3*n - 3*k + 1)A(n - 1, k - 1) + (3*k - 2)A(n - 1, k);
%F A154698 t(n,m)=(p^(n - m)*q^m + p^m*q^(n - m))*A(n+1,m+1)
%e A154698 {2},
%e A154698 {5, 5},
%e A154698 {13, 96, 13},
%e A154698 {35, 1170, 1170, 35},
%e A154698 {97, 12948, 39312, 12948, 97},
%e A154698 {275, 142170, 986760, 986760, 142170, 275},
%e A154698 {793, 1585368, 22077900, 47364480, 22077900, 1585368, 793},
%e A154698 {2315, 18009750, 470999340, 1846449000, 1846449000,
470999340, 18009750, 2315},
%e A154698 {6817, 207838956, 9861575616, 64802164752, 115218417600,
64802164752, 9861575616, 207838956, 6817},
%e A154698 {20195, 2427319170, 205220466000, 2150319921120,
6137293885920, 6137293885920, 2150319921120, 205220466000, 2427319170,
20195},
%e A154698 {60073, 28592134080, 4267189604340, 69149645568000,
298491222575520, 471344170438656, 298491222575520, 69149645568000,
4267189604340, 28592134080, 60073}
%t A154698 Clear[t, p, q, n, m, A]; A[n_, 1] := 1; A[n_, n_] := 1;
%t A154698 A[n_, k_] := (3*n - 3*k + 1)A[n - 1, k - 1] + (3*k - 2)A[n -
1, k];
%t A154698 p = 2; q = 3;
%t A154698 t[n_, m_] := (p^(n - m)*q^m + p^m*q^(n - m))*A[n + 1, m + 1];
%t A154698 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154698 Flatten[%]
%K A154698 nonn,tabl
%O A154698 0,1
%A A154698 Roger L. Bagula and Gary W. Adamson
(rlbagulatftn(AT)yahoo.com), Jan 14 2009
--
Respectfully, Roger L. Bagula
11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814
:http://www.geocities.com/rlbagulatftn/Index.html
alternative email: ***@sbcglobal.net
