A385443 Expansion of e.g.f. (1/x) * Series_Reversion( x/(3*x + sqrt(9*x^2+1))^(1/3) ).

1, 1, 3, 7, -55, -1215, -8645, 150535, 6200145, 73698625, -1986309325, -119693799225, -1993326710375, 72724743316225, 5768642653648875, 123556356142594375, -5685256808745889375, -559310285769833973375, -14644269999088713108125, 813361265343230663434375

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Index entries for reversions of series

Formula

E.g.f.: (1/x) * Series_Reversion( x * exp(-arcsinh(3*x)/3) ).

E.g.f.: ( (1/x) * Series_Reversion( x/(1 + 6*x)^(2/3) ) )^(1/4).

E.g.f. A(x) satisfies A(x) = exp( (1/3) * arcsinh(3*x*A(x)) ).

E.g.f. A(x) satisfies A(x) = (1 + 6*x*A(x)^4)^(1/6).

a(n) = 6^n * n! * binomial((4*n+1)/6,n)/(4*n+1).

a(n) = Sum_{k=0..n} (n+1)^(k-1) * (3*i)^(n-k) * A385343(n,k), where i is the imaginary unit.

Prog

(PARI) a(n) = 6^n*n!*binomial((4*n+1)/6, n)/(4*n+1);

Crossrefs

Cf. A001147, A384241, A385444.

Cf. A385343.

Sequence in context: A228490 A130294 A321968 * A100772 A320724 A259266

Adjacent sequences: A385440 A385441 A385442 * A385444 A385445 A385446

Keyword

sign

Author

Seiichi Manyama, Jun 29 2025

Status

approved