A368719 -id:A368719 - cp's OEIS frontend

A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

1, 0, 2, 0, 1, 5, 0, 1, 4, 16, 0, 1, 6, 15, 65, 0, 1, 10, 27, 64, 326, 0, 1, 18, 57, 124, 325, 1957, 0, 1, 34, 135, 292, 645, 1956, 13700, 0, 1, 66, 345, 796, 1585, 3906, 13699, 109601, 0, 1, 130, 927, 2404, 4605, 9726, 27391, 109600, 986410, 0, 1, 258, 2577, 7804, 15145, 28926, 68425, 219192, 986409, 9864101

Offset: 0

Seiichi Manyama, Aug 14 2020

			Square array begins:
     1,    0,    0,    0,     0,     0,      0, ...
     2,    1,    1,    1,     1,     1,      1, ...
     5,    4,    6,   10,    18,    34,     66, ...
    16,   15,   27,   57,   135,   345,    927, ...
    65,   64,  124,  292,   796,  2404,   7804, ...
   326,  325,  645, 1585,  4605, 15145,  54645, ...
  1957, 1956, 3906, 9726, 28926, 98646, 374526, ...

Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Columns k=0..5 give A000522, A007526, A030297, A337001, A337002, A368719.
Main diagonal gives A256016.
Cf. A368724.

T[n_, k_] := n! * Sum[If[j == k == 0, 1, j^k]/j!, {j, 0, n}]; Table[T[k, n-k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 29 2021 *)

T(0,k) = 0^k and T(n,k) = n^k + n * T(n-1,k) for n>0.

E.g.f. of column k: B_k(x) * exp(x) / (1-x), where B_n(x) = Bell polynomials. - Seiichi Manyama, Jan 04 2024

A368718 a(n) = n! * Sum_{k=0..n} (-1)^(n-k) * k^5 / k!.

Original entry on oeis.org

0, 1, 30, 153, 412, 1065, 1386, 7105, -24072, 275697, -2656970, 29387721, -352403820, 4581620953, -64142155518, 962133092145, -15394128425744, 261700184657505, -4710603321945522, 89501463119441017, -1790029262385620340, 37590614510102111241

Offset: 0

Seiichi Manyama, Jan 04 2024

sign

In general, for m >=0, Sum_{k=0..n} (-1)^(n-k) * k^m / k! ~ A000587(m) * (-1)^n * exp(-1). - Vaclav Kotesovec, Jul 18 2025

Robert Israel, Table of n, a(n) for n = 0..448
Eric Weisstein's World of Mathematics, Bell Polynomial.
Wikipedia, Touchard polynomials

Column k=5 of A368724.

Cf. A048993, A368587, A368719.

f:= proc(n) option remember;
  - n*procname(n-1)+n^5
end proc:
f(0):= 0:
seq(f(i),i=0..21); # Robert Israel, May 13 2025

Table[-5*n + 3*n^3 + n^4 - 2*(-1)^n*n*Subfactorial[n-1], {n, 0, 20}] (* Vaclav Kotesovec, Jul 18 2025 *)

my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, 5, stirling(5, k, 2)*x^k)*exp(x)/(1+x))))

a(0) = 0; a(n) = -n*a(n-1) + n^5.
E.g.f.: B_5(x) * exp(x) / (1+x), where B_n(x) = Bell polynomials.
a(n) ~ -2*(-1)^n * exp(-1) * n!. - Vaclav Kotesovec, Jul 18 2025

cp's OEIS Frontend

A337085 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} j^k/j!.

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