A377269 G.f. A(x) satisfies A(x) = (1 - 9*x*A(x))^(2/3).

1, -6, 27, -90, 189, 0, -1782, 6318, 0, -90882, 360126, 0, -5985819, 24931800, 0, -446371074, 1912892355, 0, -35840971530, 156454458930, 0, -3022929941616, 13367712796110, 0, -264079216747476, 1179032268616902, 0, -23685874363658232, 106533987128598645, 0

Offset

0,2

Links

Paolo Xausa, Table of n, a(n) for n = 0..1000

Index entries for reversions of series.

Formula

G.f.: (1/x) * Series_Reversion( x/(1-9*x)^(2/3) ).

a(n) = 9^n * binomial(n/3 - 5/3,n)/(n+1).

From Seiichi Manyama, Jun 22 2025: (Start)

G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^(1/2)).

a(3*n+2) = 0 for n > 0. (End)

E.g.f.: (27*x^2 + 2*hypergeom([-2/3, 5/6], [1/3, 2/3, 2/3, 1], 4*x^3) - 12*x*hypergeom([-1/3, 7/6], [2/3, 1, 4/3, 4/3], 4*x^3))/2. - Stefano Spezia, Jun 22 2025

D-finite with recurrence n*(n-1)*a(n) - 54*(2*n-1)*(n-5)*a(n-3) = 0. - R. J. Mathar, Jul 30 2025

a(n) ~ (-1)^(n mod 3) * 2^(2*(n+1)/3) * 3^(n+1/2) / (n^(3/2) * sqrt(Pi)) if (n mod 3) != 2. - Amiram Eldar, Nov 21 2025

Mathematica

A377269[n_] := 9^n*Binomial[(n - 5)/3, n]/(n + 1);
Array[A377269, 35, 0] (* Paolo Xausa, Aug 05 2025 *)

Prog

(PARI) a(n) = 9^n*binomial(n/3-5/3, n)/(n+1);

Crossrefs

Cf. A376636, A377268.

Sequence in context: A392444 A131985 A125196 * A100189 A052267 A038166

Adjacent sequences: A377266 A377267 A377268 * A377270 A377271 A377272

Keyword

sign,easy

Author

Seiichi Manyama, Oct 22 2024

Status

approved