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A357736 Expansion of e.g.f. sin( sqrt(2) * (exp(x) - 1) )/sqrt(2).

%I A357736 #17 Feb 16 2025 08:34:04
%S A357736 0,1,1,-1,-11,-45,-119,-49,2045,18075,105121,436471,679669,-10538333,
%T A357736 -155858247,-1404609569,-9667430739,-46708291093,-25694453615,
%U A357736 3002522206471,49051481154341,546022210068595,4800733688293929,31399017314213487,75507020603213405
%N A357736 Expansion of e.g.f. sin( sqrt(2) * (exp(x) - 1) )/sqrt(2).
%H A357736 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357736 a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^(k) * Stirling2(n,2*k+1).
%F A357736 a(n) = 0; a(n) = Sum_{k=0..n-1} binomial(n-1, k) * A357725(k).
%F A357736 a(n) = ( Bell_n(sqrt(2) * i) - Bell_n(-sqrt(2) * i) )/(2 * sqrt(2) * i), where Bell_n(x) is n-th Bell polynomial and i is the imaginary unit.
%o A357736 (PARI) my(N=30, x='x+O('x^N)); concat(0, apply(round, Vec(serlaplace(sin(sqrt(2)*(exp(x)-1))/sqrt(2)))))
%o A357736 (PARI) a(n) = sum(k=0, (n-1)\2, (-2)^k*stirling(n, 2*k+1, 2));
%o A357736 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357736 a(n) = round((Bell_poly(n, sqrt(2)*I)-Bell_poly(n, -sqrt(2)*I))/(2*sqrt(2)*I));
%Y A357736 Cf. A264037, A357725.
%K A357736 sign
%O A357736 0,5
%A A357736 _Seiichi Manyama_, Oct 11 2022