A357728 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. cos( sqrt(k) * (exp(x) - 1) ).

1, 1, 0, 1, 0, 0, 1, 0, -1, 0, 1, 0, -2, -3, 0, 1, 0, -3, -6, -6, 0, 1, 0, -4, -9, -10, -5, 0, 1, 0, -5, -12, -12, 10, 33, 0, 1, 0, -6, -15, -12, 45, 190, 266, 0, 1, 0, -7, -18, -10, 100, 465, 1106, 1309, 0, 1, 0, -8, -21, -6, 175, 852, 2394, 4438, 4905, 0, 1, 0, -9, -24, 0, 270, 1345, 4004, 7827, 9978, 11516, 0

Offset

0,13

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Eric Weisstein's World of Mathematics, Bell Polynomial.

Formula

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

T(n,k) = ( Bell_n(sqrt(k) * i) + Bell_n(-sqrt(k) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.

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, -6, -10, -12, -12, -10, ...
  0, -5,  10,  45, 100, 175, ...

Prog

(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)*I)+Bell_poly(n, -sqrt(k)*I)))/2;

Crossrefs

Columns k=0-4 give: A000007, A121867, A357725, A357726, A357727.

Main diagonal gives A357729.

Cf. A357681, A357720.

Sequence in context: A305806 A064722 A259748 * A357681 A357720 A357712

Adjacent sequences: A357725 A357726 A357727 * A357729 A357730 A357731

Keyword

sign,tabl

Author

Seiichi Manyama, Oct 11 2022

Status

approved