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A357693 Expansion of e.g.f. cos( sqrt(2) * log(1+x) ).

%I A357693 #43 Jul 23 2025 09:07:58
%S A357693 1,0,-2,6,-18,60,-216,756,-1620,-14256,349272,-5452920,78885576,
%T A357693 -1143659088,17074183104,-265437239760,4316991698448,-73572489226368,
%U A357693 1314108286270560,-24584195654596512,481215937895868384,-9843358555320333120,210128893733994567552
%N A357693 Expansion of e.g.f. cos( sqrt(2) * log(1+x) ).
%H A357693 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357693 a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling1(n,2*k).
%F A357693 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 A357693 a(0) = 1, a(1) = 0; a(n) = -(2*n-3) * a(n-1) - (n^2-4*n+6) * a(n-2).
%t A357693 With[{nn=30},CoefficientList[Series[Cos[Sqrt[2]Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Nov 04 2024 *)
%o A357693 (PARI) my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*log(1+x)))))
%o A357693 (PARI) a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 1));
%o A357693 (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 A357693 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=2, n, v[i+1]=-(2*i-3)*v[i]-(i^2-4*i+6)*v[i-1]); v;
%Y A357693 Column k=2 of A357720.
%Y A357693 Cf. A003703, A357718, A357719.
%Y A357693 Cf. A357725.
%K A357693 sign
%O A357693 0,3
%A A357693 _Seiichi Manyama_, Oct 10 2022