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

0,13

Comments

A(n,k) is the k-fold exponential convolution of A000166 with themselves, evaluated at n.

Links

Table of n, a(n) for n=0..77.

N. J. A. Sloane, Transforms

Index entries for sequences related to factorial numbers

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)

Example

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,  ...

Mathematica

Table[Function[k, n! SeriesCoefficient[Exp[-k x]/(1 - x)^k, {x, 0, n}]][j - n], {j, 0, 11}, {n, 0, j}] // Flatten

Prog

(PARI) a(n, k) = n!*sum(j=0, n, (-k)^(n-j)*binomial(j+k-1, j)/(n-j)!); \\ Seiichi Manyama, Apr 25 2025

Crossrefs

Columns k=0..5 give A000007, A000166, A087981, A137775, A383344, A383384.

Rows n=0..3 give A000012, A000004, A001477, A005843.

Main diagonal gives A295182.

Cf. A008279, A265609.

Sequence in context: A389095 A251690 A187752 * A215573 A163537 A219946

Adjacent sequences: A295178 A295179 A295180 * A295182 A295183 A295184

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Nov 16 2017

Status

approved