A368718 -id:A368718 - cp's OEIS frontend

A368724 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} (-1)^(n-j) * j^k/j!.

1, 0, 0, 0, 1, 1, 0, 1, 0, -2, 0, 1, 2, 3, 9, 0, 1, 6, 3, -8, -44, 0, 1, 14, 9, 4, 45, 265, 0, 1, 30, 39, 28, 5, -264, -1854, 0, 1, 62, 153, 100, -15, 6, 1855, 14833, 0, 1, 126, 543, 412, 125, 306, 7, -14832, -133496, 0, 1, 254, 1809, 1924, 1065, 546, -1799, 8, 133497, 1334961

Offset: 0

Seiichi Manyama, Jan 04 2024

			Square array begins:
    1,    0, 0,   0,   0,    0,     0, ...
    0,    1, 1,   1,   1,    1,     1, ...
    1,    0, 2,   6,  14,   30,    62, ...
   -2,    3, 3,   9,  39,  153,   543, ...
    9,   -8, 4,  28, 100,  412,  1924, ...
  -44,   45, 5, -15, 125, 1065,  6005, ...
  265, -264, 6, 306, 546, 1386, 10626, ...

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

Columns k=0..5 give A182386, (-1)^(n-1) * A000240(n), A001477, A368716, A368717, A368718.
Main diagonal gives A368725.
Cf. A337085.

T(n,k) = n!*sum(j=0, n, (-1)^(n-j)*j^k/j!);

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.

A368719 a(n) = n! * Sum_{k=0..n} k^5 / k!.

Original entry on oeis.org

0, 1, 34, 345, 2404, 15145, 98646, 707329, 5691400, 51281649, 512916490, 5642242441, 67707158124, 880193426905, 12322708514494, 184840628476785, 2957450056677136, 50276650964931169, 904979717370650610, 17194614630044837689, 343892292600899953780

Offset: 0

Seiichi Manyama, Jan 04 2024

nonn

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

Column k=5 of A337085.

Cf. A048993, A368718.

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) ~ 52*exp(1)*n!. - Vaclav Kotesovec, Jan 13 2024

cp's OEIS Frontend

A368724 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) = n! * Sum_{j=0..n} (-1)^(n-j) * j^k/j!.

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