A357719 Expansion of e.g.f. cos( 2 * log(1+x) ).

1, 0, -4, 12, -28, 40, 200, -3360, 35680, -357120, 3644800, -38896000, 437756800, -5206406400, 65372153600, -864339840000, 11991424640000, -173800340480000, 2617640829440000, -40693929269760000, 647089190924800000, -10383194262604800000

Offset: 0

Author

Seiichi Manyama, Oct 10 2022

Keywords

• sign

Links

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

Crossrefs

Column k=4 of A357720.

Cf. A357711, A357727.

Programs

PARI

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

PARI

a(n) = sum(k=0, n\2, (-4)^k*stirling(n, 2*k, 1));

PARI

a(n) = (-1)^n*(prod(k=0, n-1, 2*I+k)+prod(k=0, n-1, -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+8)*v[i-1]); v;

Formula

a(n) = Sum_{k=0..floor(n/2)} (-4)^k * Stirling1(n,2*k).

a(n) = (-1)^n * ( (2 * i)_n + (-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+8) * a(n-2).