This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357682 #21 Feb 16 2025 08:34:04
%S A357682 1,0,2,9,44,325,2742,24794,250168,2796795,33842610,439337085,
%T A357682 6100179780,90139379928,1409779442190,23242554452745,402652762232048,
%U A357682 7308371248274949,138605556986785674,2740167375732394378,56350604098768558140,1203156656491936711635
%N A357682 a(n) = Sum_{k=0..floor(n/2)} n^k * Stirling2(n,2*k).
%H A357682 Andrew Howroyd, <a href="/A357682/b357682.txt">Table of n, a(n) for n = 0..200</a>
%H A357682 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357682 a(n) = n! * [x^n] cosh( sqrt(n) * (exp(x) - 1) ).
%F A357682 a(n) = ( Bell_n(sqrt(n)) + Bell_n(-sqrt(n)) )/2, where Bell_n(x) is n-th Bell polynomial.
%o A357682 (PARI) a(n) = sum(k=0, n\2, n^k*stirling(n, 2*k, 2));
%o A357682 (PARI) a(n) = round(n!*polcoef(cosh(sqrt(n)*(exp(x+x*O(x^n))-1)), n));
%o A357682 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357682 a(n) = round((Bell_poly(n, sqrt(n))+Bell_poly(n, -sqrt(n))))/2;
%Y A357682 Main diagonal of A357681.
%Y A357682 Cf. A242817, A357683.
%K A357682 nonn
%O A357682 0,3
%A A357682 _Seiichi Manyama_, Oct 09 2022