A357681 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cosh( sqrt(k) * (exp(x) - 1) ).

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 8, 0, 1, 0, 4, 9, 18, 25, 0, 1, 0, 5, 12, 30, 70, 97, 0, 1, 0, 6, 15, 44, 135, 330, 434, 0, 1, 0, 7, 18, 60, 220, 705, 1694, 2095, 0, 1, 0, 8, 21, 78, 325, 1228, 3906, 9202, 10707, 0, 1, 0, 9, 24, 98, 450, 1905, 7196, 22953, 53334, 58194, 0

Offset: 0

Author

Seiichi Manyama, Oct 09 2022

Keywords

• nonn
• tabl

Examples

			Square array begins:
  1,  1,  1,   1,   1,   1, ...
  0,  0,  0,   0,   0,   0, ...
  0,  1,  2,   3,   4,   5, ...
  0,  3,  6,   9,  12,  15, ...
  0,  8, 18,  30,  44,  60, ...
  0, 25, 70, 135, 220, 325, ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Columns k=0-4 give: A000007, A024430, A264036, A357615, A065143.

Column k=9 gives A357667.

Main diagonal gives A357682.

Cf. A292860.

Programs

PARI

T(n, k) = sum(j=0, n\2, k^j*stirling(n, 2*j, 2));

PARI

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = round((Bell_poly(n, sqrt(k))+Bell_poly(n, -sqrt(k))))/2;

Formula

T(n,k) = Sum_{j=0..floor(n/2)} k^j * Stirling2(n,2*j).

T(n,k) = ( Bell_n(sqrt(k)) + Bell_n(-sqrt(k)) )/2, where Bell_n(x) is n-th Bell polynomial.