A295181 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-k*x)/(1 - x)^k.
1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 3, 4, 9, 0, 1, 0, 4, 6, 24, 44, 0, 1, 0, 5, 8, 45, 128, 265, 0, 1, 0, 6, 10, 72, 252, 880, 1854, 0, 1, 0, 7, 12, 105, 416, 1935, 6816, 14833, 0, 1, 0, 8, 14, 144, 620, 3520, 16146, 60032, 133496, 0, 1, 0, 9, 16, 189, 864, 5725, 31104, 153657, 589312, 1334961, 0
Offset: 0
Examples
E.g.f. of column k: A_k(x) = 1 + k*x^2/2! + 2*k*x^3/3! + 3*k*(k + 2)*x^4/4! + 4*k*(5*k + 6)*x^5/5! + 5*k*(3*k^2 + 26*k + 24)*x^6/6! + ... Square array begins: 1, 1, 1, 1, 1, 1, ... 0, 0, 0, 0, 0, 0, ... 0, 1, 2, 3, 4, 5, ... 0, 2, 4, 6, 8, 10, ... 0, 9, 24, 45, 72, 105, ... 0, 44, 128, 252, 416, 620, ...
Links
- N. J. A. Sloane, Transforms
- Index entries for sequences related to factorial numbers
Crossrefs
Programs
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Mathematica
Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten -
PARI
a(n, k) = n!*sum(j=0, n, (-k)^(n-j)*binomial(j+k-1, j)/(n-j)!); \\ Seiichi Manyama, Apr 25 2025
Formula
E.g.f. of column k: exp(-k*x)/(1 - x)^k.
From Seiichi Manyama, Apr 25 2025: (Start)
A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k-1,j)/(n-j)!.
A(0,k) = 1, A(1,k) = 0; A(n,k) = (n-1) * (A(n-1,k) + k*A(n-2,k)). (End)
Comments