A306189
Number of minimum dominating sets in the n-Sierpinski gasket graph.
Original entry on oeis.org
3, 6, 2, 392, 1656976026, 122836566640423857273582993856, 50043395758253154294953783566500246788902420299683914045600060272160541415159062540151890
Offset: 1
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a(n)={my(s=[[1+O(x),0,0,0],[0,0,x+O(x^2)],[0,x^2+O(x^3)],[x^3+O(x^4)]]);for(k=2,n,s=vector(4,i,vector(5-i,j,sum(xy=0,3,sum(xz=0,3,sum(yz=0,3,s[1+(i>1)+!xy+!xz][1+(j>3)+(xy%2)+(xz%2)]*s[1+(i>2)+!xy+!yz][1+(j>2)+(xy\2)+(yz%2)]*s[1+(i>3)+!xz+!yz][1+(j>1)+(xz\2)+(yz\2)]/x^(!xy+!xz+!yz)))))));pollead([1,3,3,1]*vectorv(4,i,s[i][5-i]))} \\ Christian Sievers, Jul 21 2024, improved Jul 25 2024
A377657
Array read by ascending antidiagonals: A(n, k) = Sum_{j=0..k} tan(j*Pi/(1 + 2*k))^(2*n).
Original entry on oeis.org
1, 0, 2, 0, 3, 3, 0, 9, 10, 4, 0, 27, 90, 21, 5, 0, 81, 850, 371, 36, 6, 0, 243, 8050, 7077, 1044, 55, 7, 0, 729, 76250, 135779, 33300, 2365, 78, 8, 0, 2187, 722250, 2606261, 1070244, 113311, 4654, 105, 9, 0, 6561, 6841250, 50028755, 34420356, 5476405, 312390, 8295, 136, 10
Offset: 0
Array begins
[0] 1, 2, 3, 4, 5, 6, ... A000027
[1] 0, 3, 10, 21, 36, 55, ... A014105
[2] 0, 9, 90, 371, 1044, 2365, ... A377858
[3] 0, 27, 850, 7077, 33300, 113311, ... A376778
[4] 0, 81, 8050, 135779, 1070244, 5476405, ...
[5] 0, 243, 76250, 2606261, 34420356, 264893255, ...
[6] 0, 729, 722250, 50028755, 1107069876, 12813875437, ...
[7] 0, 2187, 6841250, 960335173, 35607151476, 619859803695, ...
.
Seen as a triangle T(n, k) = A(n-k, k):
[0] 1;
[1] 0, 2;
[2] 0, 3, 3;
[3] 0, 9, 10, 4;
[4] 0, 27, 90, 21, 5;
[5] 0, 81, 850, 371, 36, 6;
[6] 0, 243, 8050, 7077, 1044, 55, 7;
[7] 0, 729, 76250, 135779, 33300, 2365, 78, 8;
[8] 0, 2187, 722250, 2606261, 1070244, 113311, 4654, 105, 9;
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A := (n, k) -> add(tan(j*Pi/(1 + 2*k))^(2*n), j = 0..k):
seq(print(seq(round(evalf(A(n, k), 32)), k = 0..6)), n = 0..7);
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A(n, k) = {trace(matcompanion(sum(m=0, k, x^m*binomial(2*k+1, 2*(k-m))*(-1)^(m+1)))^n)+(n==0) } \\ Thomas Scheuerle, Nov 11 2024
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