A381518 Expansion of e.g.f. ( (1/x) * Series_Reversion( x/(1 + sin(x))^2 ) )^(1/2).

1, 1, 4, 29, 304, 4141, 68832, 1337881, 29432576, 712263961, 18403873280, 487814777141, 12296236382208, 230142147098501, -2906327530115072, -800177574047914831, -75835523291585773568, -6054072134316123116495, -459749417224473755910144, -34556942957229166465685555

Offset: 0

Author

Seiichi Manyama, Feb 26 2025

Keywords

• sign

Links

• Index entries for reversions of series

Crossrefs

Cf. A136630, A198865, A381519.

Programs

PARI

a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a(n) = sum(k=0, n, k!*binomial(2*n+1, k)*I^(n-k)*a136630(n, k))/(2*n+1);

Formula

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

a(n) = (1/(2*n+1)) * Sum_{k=0..n} k! * binomial(2*n+1,k) * i^(n-k) * A136630(n,k), where i is the imaginary unit.