A393660 Expansion of (1/x) * Series_Reversion( x * ( Sum_{k=0..5} (-x)^k )^3 ).

1, 3, 12, 55, 273, 1428, 7755, 43335, 247728, 1442895, 8537400, 51198084, 310618501, 1903704213, 11771483559, 73361346135, 460383173145, 2907060507780, 18457935937675, 117776038227135, 754846049949372, 4857336644267848, 31369745876798157, 203259995090830620

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Formula

G.f.: (1/x) * Series_Reversion( x * ((1-x^6) / (1+x))^3 ).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/6)} binomial(3*n+k+2,k) * binomial(3*n+3,n-6*k).

Mathematica

CoefficientList[Normal@Series[InverseSeries@Series[x*((1-x^6)/(1+x))^3, {x, 0, 50}]/x, {x, 0, 20}], x] (* Vincenzo Librandi, Mar 24 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\6, binomial(3*n+k+2, k)*binomial(3*n+3, n-6*k))/(n+1);
(Magma) N := 25; R<x> := PowerSeriesRing(Rationals(), N+5); f:= x*((1-x^6)/(1+x))^3; g:= Reverse(f) div x; Coeffs := [Coefficient(g, i):i in [0..N]]; Coeffs; // Vincenzo Librandi, Mar 24 2026

Crossrefs

Cf. A393658, A393659.

Sequence in context: A007199 A179848 A001764 * A171780 A369159 A216493

Adjacent sequences: A393657 A393658 A393659 * A393661 A393662 A393663

Keyword

nonn

Author

Seiichi Manyama, Feb 24 2026

Status

approved