A381519 Expansion of e.g.f. (1/x) * Series_Reversion( x/(1 + sin(x))^2 ).

1, 2, 10, 82, 936, 13642, 240656, 4952218, 115608704, 2992207250, 84070140672, 2507383885730, 77117178496000, 2329071118971482, 61202811821836288, 690380688651775978, -88097620429234470912, -11900508444760552311518, -1112180862634722333884416

Offset: 0

Author

Seiichi Manyama, Feb 26 2025

Keywords

• sign

Links

• Index entries for reversions of series

Crossrefs

Cf. A136630, A381389, A381518.

Cf. A377553, A381442, A381449, A381521.

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+2, k)*I^(n-k)*a136630(n, k))/(n+1);

Formula

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

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A381518.

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