A381520 Expansion of e.g.f. ( (1/x) * Series_Reversion( x/(1 + x * cos(x))^2 ) )^(1/2).

1, 1, 4, 27, 240, 2345, 17280, -226597, -21007616, -1007159823, -42976972800, -1775328986981, -72123329507328, -2843431148886887, -103621659777126400, -2971936506262036965, -6719764584265482240, 9528526268302653725537, 1192610999728818101551104

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, A381378, A381447, A381479.

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+1, k)*I^(n-k)*a185951(n, k))/(2*n+1);

Formula

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

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