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 A357725 #24 Feb 16 2025 08:34:04
%S A357725 1,0,-2,-6,-10,10,190,1106,4438,9978,-35250,-666622,-5657370,
%T A357725 -35308182,-155215970,-128513870,7051468022,105057922906,
%U A357725 1042016038254,8053738122466,44608555196294,48639210067658,-3200193654245442,-60669816166988654,-769281697485061994
%N A357725 Expansion of e.g.f. cos( sqrt(2) * (exp(x) - 1) ).
%H A357725 Andrew Howroyd, <a href="/A357725/b357725.txt">Table of n, a(n) for n = 0..200</a>
%H A357725 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357725 a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling2(n,2*k).
%F A357725 a(n) = 1; a(n) = -2 * Sum_{k=0..n-1} binomial(n-1, k) * A357736(k).
%F A357725 a(n) = ( Bell_n(sqrt(2) * i) + Bell_n(-sqrt(2) * i) )/2, where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.
%o A357725 (PARI) my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*(exp(x)-1)))))
%o A357725 (PARI) a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 2));
%o A357725 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357725 a(n) = round((Bell_poly(n, sqrt(2)*I)+Bell_poly(n, -sqrt(2)*I)))/2;
%Y A357725 Column k=2 of A357728.
%Y A357725 Cf. A121867, A264036, A357736.
%K A357725 sign
%O A357725 0,3
%A A357725 _Seiichi Manyama_, Oct 10 2022