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%I #37 Oct 12 2022 08:58:42
%S 1,0,-2,6,-18,60,-216,756,-1620,-14256,349272,-5452920,78885576,
%T -1143659088,17074183104,-265437239760,4316991698448,-73572489226368,
%U 1314108286270560,-24584195654596512,481215937895868384,-9843358555320333120,210128893733994567552
%N Expansion of e.g.f. cos( sqrt(2) * log(1+x) ).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling1(n,2*k).
%F a(n) = (-1)^n * ( (sqrt(2) * i)_n + (-sqrt(2) * i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit.
%F a(0) = 1, a(1) = 0; a(n) = -(2*n-3) * a(n-1) - (n^2-4*n+6) * a(n-2).
%o (PARI) my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*log(1+x)))))
%o (PARI) a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 1));
%o (PARI) a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(2)*I+k)+prod(k=0, n-1, -sqrt(2)*I+k)))/2;
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; v[2]=0; for(i=2, n, v[i+1]=-(2*i-3)*v[i]-(i^2-4*i+6)*v[i-1]); v;
%Y Column k=2 of A357720.
%Y Cf. A003703, A357718, A357719.
%Y Cf. A357725.
%K sign
%O 0,3
%A _Seiichi Manyama_, Oct 10 2022
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