pictures of the fractals

[Roger Bagula]

The {2,1} levels: Pascal and Eulerian:Modulo 3

Clear[t, p, q, n, m, a]
p = 2; q = 1;
t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*Binomial[n, m]
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]
Flatten[%]
Table[Sum[t[n, m], {m, 0, n}], {n, 0, 10}]
a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 32}];
b = Table[If[m ≤ n, Mod[a[[
n]][[m]], 3], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];
ListDensityPlot[b, Mesh -> False]

Loading Image...

Clear[t, p, q, n, m, a]
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}]
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]
Flatten[%]
Table[Sum[t[n, m], {m, 0, n}], {n, 0, 10}]
a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 32}];
b = Table[If[m ≤ n, Mod[a[[n]][[m]], 3], 0], {m, 1, Length[a]}, {
n, 1, Length[a]}];
ListDensityPlot[b, Mesh -> False]

Loading Image...

The {2,3} levels: Pascal and Eulerian:Modulo 5

Clear[t, p, q, n, m]
p = 2; q = 3;
t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*Binomial[n, m]
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]
Flatten[%]
Table[Sum[t[n, m], {m, 0, n}], {n, 0, 10}]
a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 32}];
b = Table[If[m ≤ n, Mod[
a[[n]][[m]], 5], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];
ListDensityPlot[b, Mesh -> False]

Loading Image...

Clear[t, p, q, n, m]
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}]
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]
Flatten[%]
Table[Sum[t[n, m], {m, 0, n}], {n, 0, 10}]
a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 32}];
b = Table[If[m ≤ n, Mod[a[[n]][[m]], 5], 0], {m, 1, Length[a]}, {n, 1,
Length[a]}];
ListDensityPlot[b, Mesh -> False]

Loading Image...

[Roger Bagula]

New symmetrical Stirling's types and a Pascal scale 3:
Pascal {2,1}
Clear[t, p, q, n, m, a]
p = 2; q = 1;
t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*Binomial[n, m]
a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 128}];
b = Table[If[m ≤ n, 3 - Mod[a[[n]][[m]], 3], 0], {m,
1, Length[a]}, {n, 1, Length[a]}];
ListDensityPlot[b, Mesh -> False, Frame -> False]
Loading Image...

Striling type1 symmetrical:
Clear[t, n, m]
t[n_, m_] = StirlingS1[n, m] + StirlingS1[n, n - m]
a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 128}];
b = Table[If[m ≤ n, 1 - Mod[
a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];
ListDensityPlot[b, Mesh -> False, Frame -> False]
Loading Image...

Stirling type 2 symmetrical:
Clear[t, n, m]
t[n_, m_] = StirlingS2[n, m] + StirlingS2[n, n - m]
a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 128}];
b = Table[If[m ≤ n, 1 - Mod[a[[n]][[m]], 2], 0], {m,
1, Length[a]}, {n, 1, Length[a]}];
ListDensityPlot[b, Mesh -> False, Frame -> False]
Loading Image...

[Roger Bagula]

Last night I thought of applying the generalized Sierpinsi-Pascal
combinations to the new symmetrical Stirling numbers.
The Stirling second kind give some pretty fractals.
It is sort of puzzling where Stirling numbers fit in:
it has to be between Pascal and Eulerian numbers.


%I A154913
%S A154913 4,3,3,5,8,5,9,6,6,9,17,120,176,120,17,33,252,180,180,252,33,65,
%T A154913 4590,7180,7200,7180,4590,65,129,46134,57204,21336,21336,57204,
%U A154913
46134,129,257,658840,910520,603680,433216,603680,910520,658840,257
%V A154913
4,3,3,5,-8,5,9,-6,-6,9,17,-120,176,-120,17,33,252,-180,-180,252,33,65,
%W A154913
-4590,7180,-7200,7180,-4590,65,129,46134,-57204,21336,21336,-57204,
%X A154913
46134,129,257,-658840,910520,-603680,433216,-603680,910520,-658840,257
%N A154913 A triangular sequence: p = 2; q = 1; t(n,m) = (p^(n - m)*q^m
+ p^m*q^( n - m))*(StirlingS1[n, m] + StirlingS1[n, n - m]).
%C A154913 Row sums are:
%C A154913 {4, 6, 2, 6, -30, 210, -1890, 20790, -270270, 4054050,
-68918850,..}.
%C A154913 Fractal Plot:
%C A154913 a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 243}];
%C A154913 b = Table[If[m ≤ n, 3 - Mod[a[[n]][[m]], 3], 0], {m, 1,
Length[a]}, {n, 1, Length[a]}];
%C A154913 ListDensityPlot[b, Mesh -> False, Frame -> False,
AspectRatio -> Automatic, ColorFunction -> (Hue[2#] &)]
%F A154913 p = 2; q = 1;
%F A154913 t(n,m) = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS1[n, m] +
StirlingS1[n, n - m]).
%e A154913 {4},
%e A154913 {3, 3},
%e A154913 {5, -8, 5},
%e A154913 {9, -6, -6, 9},
%e A154913 {17, -120, 176, -120, 17},
%e A154913 {33, 252, -180, -180, 252, 33},
%e A154913 {65, -4590, 7180, -7200, 7180, -4590, 65},
%e A154913 {129, 46134, -57204, 21336, 21336, -57204, 46134, 129},
%e A154913 {257, -658840, 910520, -603680, 433216, -603680, 910520,
-658840, 257},
%e A154913 {513, 10393272, -14393016, 8178336, -2152080, -2152080,
8178336, -14393016, 10393272, 513},
%e A154913 {1025, -186543450, 267135960, -160772400, 62956240,
-34473600, 62956240, -160772400, 267135960, -186543450, 1025}
%t A154913 Clear[t, p, q, n, m, a];
%t A154913 p = 2; q = 1;
%t A154913 t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS1[n, m]
+ StirlingS1[n, n - m]);
%t A154913 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154913 Flatten[%]
%K A154913 nonn
%O A154913 0,1
%A A154913 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 17 2009

%I A154914
%S A154914 4,5,5,13,24,13,35,30,30,35,97,936,1584,936,97,275,2940,2700,
%T A154914 2700,2940,275,793,78570,168012,194400,168012,78570,793,2315,
%U A154914 1153350,2002140,960120,960120,2002140,1153350,2315,6817,24113544
%V A154914
4,5,5,13,-24,13,35,-30,-30,35,97,-936,1584,-936,97,275,2940,-2700,
%W A154914 -2700,2940,275,793,-78570,168012,-194400,168012,-78570,793,2315,
%X A154914
1153350,-2002140,960120,960120,-2002140,1153350,2315,6817,-24113544
%N A154914 A triangular sequence: p = 2; q = 3; t(n,m) = (p^(n - m)*q^m
+ p^m*q^( n - m))*(StirlingS1[n, m] + StirlingS1[n, n - m]).
%C A154914 Row sums are:
%C A154914 {4, 10, 2, 10, -94, 1030, -13930, 227290, -4363870, 96566470,
-2422269850,..}.
%C A154914 Fractal Plot:
%C A154914 a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 243}];
%C A154914 b = Table[If[m ≤ n, 5 - Mod[a[[n]][[m]], 5], 0], {m, 1,
Length[a]}, {n, 1, Length[a]}];
%C A154914 ListDensityPlot[b, Mesh -> False, Frame -> False,
AspectRatio -> Automatic, ColorFunction -> (Hue[2#] &)]
%F A154914 p = 2; q = 3;
%F A154914 t(n,m) = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS1[n, m] +
StirlingS1[n, n - m]).
%e A154914 {4},
%e A154914 {5, 5},
%e A154914 {13, -24, 13},
%e A154914 {35, -30, -30, 35},
%e A154914 {97, -936, 1584, -936, 97},
%e A154914 {275, 2940, -2700, -2700, 2940, 275},
%e A154914 {793, -78570, 168012, -194400, 168012, -78570, 793},
%e A154914 {2315, 1153350, -2002140, 960120, 960120, -2002140, 1153350,
2315},
%e A154914 {6817, -24113544, 46757880, -42378336, 35090496, -42378336,
46757880, -24113544, 6817},
%e A154914 {20195, 559544760, -1079476200, 858725280, -290530800,
-290530800, 858725280, -1079476200, 559544760, 20195},
%e A154914 {60073, -14844358350, 29331528408, -24768406800, 13258584144,
-8377084800, 13258584144, -24768406800, 29331528408, -14844358350, 60073}
%t A154914 Clear[t, p, q, n, m, a];
%t A154914 p = 2; q = 3;
%t A154914 t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS1[n, m]
+ StirlingS1[n, n - m]);
%t A154914 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154914 Flatten[%]
%K A154914 nonn
%O A154914 0,1

%I A154915
%S A154915
4,3,3,5,8,5,9,24,24,9,17,70,112,70,17,33,198,480,480,198,33,65,544,
%T A154915
1920,2880,1920,544,65,129,1452,7308,15624,15624,7308,1452,129,257,3770,
%U A154915
26724,80640,108864,80640,26724,3770,257,513,9546,94644,408312,706608
%N A154915 A triangular sequence: p = 2; q = 1; t(n,m) = (p^(n - m)*q^m
+ p^m*q^( n - m))*(StirlingS2[n, m] + StirlingS2[n, n - m]).
%C A154915 Row sums are:
%C A154915 {4, 6, 18, 66, 286, 1422, 7938, 49026, 331646, 2439246,
19394498,..}.
%C A154915 Fractal Plot:
%C A154915 a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 243}];
%C A154915 b = Table[If[m ≤ n, 3 - Mod[a[[n]][[m]], 3], 0], {m, 1,
Length[a]}, {n, 1, Length[a]}];
%C A154915 ListDensityPlot[b, Mesh -> False, Frame -> False,
AspectRatio -> Automatic, ColorFunction -> (Hue[2#] &)]
%F A154915 p = 2; q = 1;
%F A154915 t(n,m) = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS2[n, m] +
StirlingS2[n, n - m]).
%e A154915 {4},
%e A154915 {3, 3},
%e A154915 {5, 8, 5},
%e A154915 {9, 24, 24, 9},
%e A154915 {17, 70, 112, 70, 17},
%e A154915 {33, 198, 480, 480, 198, 33},
%e A154915 {65, 544, 1920, 2880, 1920, 544, 65},
%e A154915 {129, 1452, 7308, 15624, 15624, 7308, 1452, 129},
%e A154915 {257, 3770, 26724, 80640, 108864, 80640, 26724, 3770, 257},
%e A154915 {513, 9546, 94644, 408312, 706608, 706608, 408312, 94644,
9546, 513},
%e A154915 {1025, 23644, 327860, 2068560, 4554560, 5443200, 4554560,
2068560, 327860, 23644, 1025}
%t A154915 Clear[t, p, q, n, m, a];
%t A154915 p = 2; q = 1;
%t A154915 t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS2[n, m]
+ StirlingS2[n, n - m]);
%t A154915 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154915 Flatten[%]
%K A154915 nonn,tabf
%O A154915 0,1
%A A154915 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 17 2009

%I A154916
%S A154916
4,5,5,13,24,13,35,120,120,35,97,546,1008,546,97,275,2310,7200,7200,
%T A154916
2310,275,793,9312,44928,77760,44928,9312,793,2315,36300,255780,703080,
%U A154916
703080,255780,36300,2315,6817,137982,1372356,5660928,8817984,5660928
%N A154916 A triangular sequence: p = 2; q = 3; t(n,m) = (p^(n - m)*q^m
+ p^m*q^( n - m))*(StirlingS2[n, m] + StirlingS2[n, n - m]).
%C A154916 Row sums are:
%C A154916 {4, 10, 50, 310, 2294, 19570, 187826, 1994950, 23174150,
291794530, 3954319298,..}.
%C A154916 Fractal Plot:
%C A154916 a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 243}];
%C A154916 b = Table[If[m ≤ n, 5 - Mod[a[[n]][[m]], 5], 0], {m, 1,
Length[a]}, {n, 1, Length[a]}];
%C A154916 ListDensityPlot[b, Mesh -> False, Frame -> False,
AspectRatio -> Automatic, ColorFunction -> (Hue[2#] &)]
%F A154916 p = 2; q = 3;
%F A154916 t(n,m) = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS2[n, m] +
StirlingS2[n, n - m]).
%e A154916 {4},
%e A154916 {5, 5},
%e A154916 {13, 24, 13},
%e A154916 {35, 120, 120, 35},
%e A154916 {97, 546, 1008, 546, 97},
%e A154916 {275, 2310, 7200, 7200, 2310, 275},
%e A154916 {793, 9312, 44928, 77760, 44928, 9312, 793},
%e A154916 {2315, 36300, 255780, 703080, 703080, 255780, 36300, 2315},
%e A154916 {6817, 137982, 1372356, 5660928, 8817984, 5660928, 1372356,
137982, 6817},
%e A154916 {20195, 513930, 7098300, 42872760, 95392080, 95392080,
42872760, 7098300, 513930, 20195},
%e A154916 {60073, 1881492, 35999028, 318679920, 959190336, 1322697600,
959190336, 318679920, 35999028, 1881492, 60073}
%t A154916 Clear[t, p, q, n, m, a];
%t A154916 p = 2; q = 3;
%t A154916 t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS2[n, m]
+ StirlingS2[n, n - m]);
%t A154916 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154916 Flatten[%]
%K A154916 nonn,tabl
%O A154916 0,1
%A A154916 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 17 2009

[Roger Bagula]

Loading Image...
Loading Image...

%I A154922
%S A154922
4,7,7,29,40,29,133,280,280,133,641,2030,2800,2030,641,3157,14630,28000,
%T A154922
28000,14630,3157,15689,102560,278400,360000,278400,102560,15689,78253,
%U A154922
694540,2699900,4557000,4557000,2699900,694540,78253,390881,4549810
%N A154922 A triangular sequence: p = 2; q = 5; t(n,m) = (p^(n - m)*q^m
+ p^m*q^( n - m))*(StirlingS2[n, m] + StirlingS2[n, n - m]).
%C A154922 Row sums are:
%C A154922 {4, 14, 98, 826, 8142, 91574, 1153298, 16059386, 245231982,
4083954294,
%C A154922 73865689618}
%F A154922 p = 2; q = 5;
%F A154922 t(n,m) = (p^(n - m)*q^m + p^m*q^( n - m))*(StirlingS2[n, m] +
StirlingS2[n, n - m]).
%e A154922 {4},
%e A154922 {7, 7},
%e A154922 {29, 40, 29},
%e A154922 {133, 280, 280, 133},
%e A154922 {641, 2030, 2800, 2030, 641},
%e A154922 {3157, 14630, 28000, 28000, 14630, 3157}, {15689, 102560,
278400, 360000, 278400, 102560, 15689},
%e A154922 {78253, 694540, 2699900, 4557000, 4557000, 2699900, 694540,
78253},
%e A154922 {390881, 4549810, 25191300, 58464000, 68040000, 58464000,
25191300, 4549810, 390881},
%e A154922 {1953637, 28953610, 226356900, 754243000, 1030470000,
1030470000, 754243000, 226356900, 28953610, 1953637},
%e A154922 {9766649, 179805260, 1978382900, 9749610000, 16510280000,
17010000000, 16510280000, 9749610000, 1978382900, 179805260, 9766649}
%t A154922 Clear[t, p, q, n, m, a];
%t A154922 p = 2; q = 5;
%t A154922 t[n_, m_] = (p^(n - m)*q^m + p^m*q^(n - m))*(StirlingS2[n, m]
+ StirlingS2[n, n - m]);
%t A154922 Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
%t A154922 Flatten[%]
%K A154922 nonn
%O A154922 0,1
%A A154922 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 17 2009