A357725 Expansion of e.g.f. cos( sqrt(2) * (exp(x) - 1) ).
1, 0, -2, -6, -10, 10, 190, 1106, 4438, 9978, -35250, -666622, -5657370, -35308182, -155215970, -128513870, 7051468022, 105057922906, 1042016038254, 8053738122466, 44608555196294, 48639210067658, -3200193654245442, -60669816166988654, -769281697485061994
Offset: 0
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- Eric Weisstein's World of Mathematics, Bell Polynomial.
Programs
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PARI
my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*(exp(x)-1))))) -
PARI
a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 2));
-
PARI
Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!); a(n) = round((Bell_poly(n, sqrt(2)*I)+Bell_poly(n, -sqrt(2)*I)))/2;
Formula
a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling2(n,2*k).
a(n) = 1; a(n) = -2 * Sum_{k=0..n-1} binomial(n-1, k) * A357736(k).
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.