A368759 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * (1 + Sum_{j=0..n} j^k/j!).

2, 1, 3, 1, 2, 7, 1, 2, 6, 22, 1, 2, 8, 21, 89, 1, 2, 12, 33, 88, 446, 1, 2, 20, 63, 148, 445, 2677, 1, 2, 36, 141, 316, 765, 2676, 18740, 1, 2, 68, 351, 820, 1705, 4626, 18739, 149921, 1, 2, 132, 933, 2428, 4725, 10446, 32431, 149920, 1349290, 1, 2, 260, 2583, 7828, 15265, 29646, 73465, 259512, 1349289, 13492901

Offset

0,1

Links

Table of n, a(n) for n=0..65.

Eric Weisstein's World of Mathematics, Bell Polynomial.

Wikipedia, Touchard polynomials.

Formula

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

T(n,k) = n! + A337085(n,k).

E.g.f. of column k: (1+ B_k(x) * exp(x)) / (1-x), where B_n(x) = Bell polynomials.

Example

Square array begins:
     2,    1,    1,     1,     1,     1,      1, ...
     3,    2,    2,     2,     2,     2,      2, ...
     7,    6,    8,    12,    20,    36,     68, ...
    22,   21,   33,    63,   141,   351,    933, ...
    89,   88,  148,   316,   820,  2428,   7828, ...
   446,  445,  765,  1705,  4725, 15265,  54765, ...
  2677, 2676, 4626, 10446, 29646, 99366, 375246, ...

Prog

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

Crossrefs

Columns k=0..3 give A038159, A033540(n+1), A053817, A368760.

Cf. A337085.

Sequence in context: A168017 A293980 A365785 * A240694 A258643 A377009

Adjacent sequences: A368756 A368757 A368758 * A368760 A368761 A368762

Keyword

nonn,tabl

Author

Seiichi Manyama, Jan 04 2024

Status

approved