A357718 Expansion of e.g.f. cos( sqrt(3) * log(1+x) ).

1, 0, -3, 9, -24, 60, -84, -756, 13104, -157248, 1795248, -20900880, 254007936, -3250473408, 43922668608, -626830626240, 9437477107968, -149644407564288, 2493958878657792, -43592393744250624, 797394015216175104, -15230735270523601920

Offset: 0

Author

Seiichi Manyama, Oct 10 2022

Keywords

• sign

Links

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

Crossrefs

Column k=3 of A357720.

Cf. A357703, A357726.

Programs

PARI

my(N=30, x='x+O('x^N)); apply(round, Vec(serlaplace(cos(sqrt(3)*log(1+x)))))

PARI

a(n) = sum(k=0, n\2, (-3)^k*stirling(n, 2*k, 1));

PARI

a(n) = (-1)^n*round((prod(k=0, n-1, sqrt(3)*I+k)+prod(k=0, n-1, -sqrt(3)*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+7)*v[i-1]); v;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-3)^k * Stirling1(n,2*k).

a(n) = (-1)^n * ( (sqrt(3) * i)_n + (-sqrt(3) * 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+7) * a(n-2).