A385440 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x * A(x)^2) ).

1, 1, 5, 48, 693, 13440, 328185, 9676800, 334639305, 13284311040, 595505854125, 29756856729600, 1640160546688125, 98860780014796800, 6469121228247302625, 456736803668361216000, 34607895888408878660625, 2801319062499282124800000, 241247999301688986945463125

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..352

Formula

E.g.f. A(x) satisfies A(x) = (1 + 2*x*A(x)^3)^(1/2).

a(n) = 2^n * n! * binomial((3*n+1)/2,n)/(3*n+1).

a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.

a(n) ~ 3^(3*n/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Jul 04 2025

From Seiichi Manyama, Mar 22 2026: (Start)

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

E.g.f. A(x) satisfies A(x) = 1/sqrt(1 - 2*x*A(x)). (End)

D-finite with recurrence 3*(3*n + 5)*(3*n + 1)*a(n)*(n + 1) - a(n + 2)*(n + 3) = 0. - Robert Israel, Jun 10 2026

Maple

f:= gfun:-rectoproc({3*(3*n + 5)*(3*n + 1)*a(n)*(n + 1) - a(n + 2)*(n + 3), a(0)=1, a(1)=1}, a(n), remember):
map(f, [$0..20]); # Robert Israel, Jun 10 2026

Prog

(PARI) a(n) = 2^n*n!*binomial((3*n+1)/2, n)/(3*n+1);

Crossrefs

Cf. A001147, A385369, A385441, A385442.

Cf. A381415, A385343.

Sequence in context: A126224 A360235 A381417 * A380042 A108207 A381263

Adjacent sequences: A385437 A385438 A385439 * A385441 A385442 A385443

Keyword

nonn,changed

Author

Seiichi Manyama, Jun 29 2025

Status

approved