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
Examples
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, ...
Links
- Eric Weisstein's World of Mathematics, Bell Polynomial.
- Wikipedia, Touchard polynomials.
Programs
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PARI
T(n, k) = n!*(1+sum(j=0, n, j^k/j!));
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.