%I #20 Jan 22 2024 13:01:53

%S 1,0,2,9,44,325,2742,24794,250168,2796795,33842610,439337085,

%T 6100179780,90139379928,1409779442190,23242554452745,402652762232048,

%U 7308371248274949,138605556986785674,2740167375732394378,56350604098768558140,1203156656491936711635

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

%H Andrew Howroyd, <a href="/A357682/b357682.txt">Table of n, a(n) for n = 0..200</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.

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

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

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

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

%o (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);

%o a(n) = round((Bell_poly(n, sqrt(n))+Bell_poly(n, -sqrt(n))))/2;

%Y Main diagonal of A357681.

%Y Cf. A242817, A357683.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 09 2022