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

1, 0, -2, 6, -18, 60, -216, 756, -1620, -14256, 349272, -5452920, 78885576, -1143659088, 17074183104, -265437239760, 4316991698448, -73572489226368, 1314108286270560, -24584195654596512, 481215937895868384, -9843358555320333120, 210128893733994567552

Offset: 0

Author

Seiichi Manyama, Oct 10 2022

Keywords

• sign

Links

• Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Crossrefs

Column k=2 of A357720.

Cf. A003703, A357718, A357719.

Cf. A357725.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Cos[Sqrt[2]Log[1+x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 04 2024 *)

PARI

my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(2)*log(1+x)))))

PARI

a(n) = sum(k=0, n\2, (-2)^k*stirling(n, 2*k, 1));

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;

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;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-2)^k * Stirling1(n,2*k).

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.

a(0) = 1, a(1) = 0; a(n) = -(2*n-3) * a(n-1) - (n^2-4*n+6) * a(n-2).