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A256894
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Triangle read by rows, T(n,k) = Sum_{j=0..n-k+1} C(n-1,j-1)*T(n-j,k-1) if k != 0 else 1, n>=0, 0<=k<=n.
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3
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1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 13, 7, 1, 1, 16, 40, 35, 11, 1, 1, 32, 121, 155, 80, 16, 1, 1, 64, 364, 651, 490, 161, 22, 1, 1, 128, 1093, 2667, 2751, 1316, 294, 29, 1, 1, 256, 3280, 10795, 14721, 9597, 3108, 498, 37, 1, 1, 512, 9841, 43435, 76630, 65352
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OFFSET
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0,5
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COMMENTS
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Can be understood as a convolution matrix or as a sequence-to-triangle transformation similar to the partial Bell polynomials defined as: S -> T(n, k, S) = Sum_{j=0..n-k+1} C(n-1,j-1)*S(j)*T(n-j,k-1,S) if k != 0 else S(0)^n. Here S(n) = 1 for all n. The case S(n) = n gives the triangle of idempotent numbers A059297.
Conjecture: T(n,k) is the sum of two consecutive Stirling numbers of the second kind: T(n,k) = S2(n,k) + S2(n,k+1), see A008277. Checked up to n = 500. - Manfred Boergens, May 13 2024
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LINKS
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EXAMPLE
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Triangle starts:
1;
1, 1;
1, 2, 1;
1, 4, 4, 1;
1, 8, 13, 7, 1;
1, 16, 40, 35, 11, 1;
1, 32, 121, 155, 80, 16, 1;
1, 64, 364, 651, 490, 161, 22, 1;
The signed version is the inverse of A326326:
1;
-1, 1;
1, -2, 1;
-1, 4, -4, 1;
1, -8, 13, -7, 1;
-1, 16, -40, 35, -11, 1;
1, -32, 121, -155, 80, -16, 1;
-1, 64, -364, 651, -490, 161, -22, 1. - Peter Luschny, Jul 02 2019
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MAPLE
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# Implemented as a sequence transformation acting on f: n -> 1, 1, 1, 1, ... .
F := proc(n, k, f) option remember; `if`(k=0, f(0)^n,
add(binomial(n-1, j-1)*f(j)*F(n-j, k-1, f), j=0..n-k+1)) end:
for n from 0 to 7 do seq(F(n, k, j->1), k=0..n) od;
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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