A381521 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + x * cos(x))^2 ).

1, 2, 10, 78, 792, 9250, 106080, 636286, -30646784, -2237508990, -112000654080, -5124930562642, -227068649702400, -9819508698442846, -406371251899045888, -15094508095346343330, -394372545425757634560, 7096803535075158290434, 2430273114806112504446976

Offset: 0

Author

Seiichi Manyama, Feb 26 2025

Keywords

• sign

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

• Index entries for reversions of series

Crossrefs

Cf. A185951, A381480, A381520.

Cf. A377553, A381442, A381449, A381519.

Programs

PARI

a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+2, k)*I^(n-k)*a185951(n, k))/(n+1);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x) * cos(x*A(x)))^2.

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381520.

a(n) = (1/(n+1)) * Sum_{k=0..n} k! * binomial(2*n+2,k) * i^(n-k) * A185951(n,k), where i is the imaginary unit.