A357712 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) * log(1-x) ).

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 2, 3, 0, 1, 0, 3, 6, 12, 0, 1, 0, 4, 9, 26, 60, 0, 1, 0, 5, 12, 42, 140, 360, 0, 1, 0, 6, 15, 60, 240, 896, 2520, 0, 1, 0, 7, 18, 80, 360, 1614, 6636, 20160, 0, 1, 0, 8, 21, 102, 500, 2520, 12474, 55804, 181440, 0, 1, 0, 9, 24, 126, 660, 3620, 20160, 108900, 525168, 1814400, 0

Offset

0,13

Links

Table of n, a(n) for n=0..77.

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Formula

T(n,k) = Sum_{j=0..floor(n/2)} k^j * |Stirling1(n,2*j)|.

T(n,k) = ( (sqrt(k))_n + (-sqrt(k))_n )/2, where (x)_n is the Pochhammer symbol.

T(0,k) = 1, T(1,k) = 0; T(n,k) = (2*n-3) * T(n-1,k) - (n^2-4*n+4-k) * T(n-2,k).

Example

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, 12,  26,  42,  60,  80, ...
  0, 60, 140, 240, 360, 500, ...

Prog

(PARI) T(n, k) = sum(j=0, n\2, k^j*abs(stirling(n, 2*j, 1)));
(PARI) T(n, k) = round((prod(j=0, n-1, sqrt(k)+j)+prod(j=0, n-1, -sqrt(k)+j)))/2;

Crossrefs

Columns k=0-4 give: A000007, (-1)^n * A105752(n), A263687, A357703, A357711.

Main diagonal gives A357683.

Cf. A357681.

Sequence in context: A357728 A357681 A357720 * A298159 A123735 A155839

Adjacent sequences: A357709 A357710 A357711 * A357713 A357714 A357715

Keyword

nonn,tabl

Author

Seiichi Manyama, Oct 10 2022

Status

approved