A357682 a(n) = Sum_{k=0..floor(n/2)} n^k * Stirling2(n,2*k).

1, 0, 2, 9, 44, 325, 2742, 24794, 250168, 2796795, 33842610, 439337085, 6100179780, 90139379928, 1409779442190, 23242554452745, 402652762232048, 7308371248274949, 138605556986785674, 2740167375732394378, 56350604098768558140, 1203156656491936711635

Offset: 0

Author

Seiichi Manyama, Oct 09 2022

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Bell Polynomial.

Crossrefs

Main diagonal of A357681.

Cf. A242817, A357683.

Programs

PARI

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

PARI

a(n) = round(n!*polcoef(cosh(sqrt(n)*(exp(x+x*O(x^n))-1)), n));

PARI

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

Formula

a(n) = n! * [x^n] cosh( sqrt(n) * (exp(x) - 1) ).

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