A381389 -id:A381389 - cp's OEIS frontend

A381388 E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)^2) ).

1, 1, 6, 71, 1280, 31201, 961184, 35838991, 1569696768, 79007365921, 4494170889472, 285130996517399, 19963494971809792, 1529055924661457921, 127179971644212387840, 11416028319985437309215, 1099976414821996358795264, 113239907265894992879189185, 12404749306625020735299780608

Offset: 0

Seiichi Manyama, Feb 22 2025

nonn

Cf. A136630, A381389.

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

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

E.g.f.: ( (1/x) * Series_Reversion( x*(1 - sin(x))^2 ) )^(1/2).

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Feb 26 2025

sign

Index entries for reversions of series

Cf. A136630, A381389, A381518.

Cf. A377553, A381442, A381449, A381521.

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);

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.

cp's OEIS Frontend

A381388 E.g.f. A(x) satisfies A(x) = 1/( 1 - sin(x * A(x)^2) ).

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