A393659 Expansion of (1/x) * Series_Reversion( x * ( Sum_{k=0..5} (-x)^k )^2 ).

1, 2, 5, 14, 42, 132, 431, 1462, 5168, 19076, 73416, 293020, 1203387, 5043332, 21414180, 91618950, 393524230, 1693197996, 7289880565, 31394975100, 135262566139, 583216808972, 2517758495665, 10887310315980, 47174433765615, 204871463967102, 891872023961954

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))^2 ).

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

Mathematica

CoefficientList[Normal@Series[InverseSeries@Series[x*((1-x^6)/(1+x))^2, {x, 0, 50}]/x, {x, 0, 27}], x] (* Vincenzo Librandi, Mar 26 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\6, binomial(2*n+k+1, k)*binomial(2*n+2, n-6*k))/(n+1);
(Magma) N := 35; R<x> := PowerSeriesRing(Rationals(), N+5); f:= x*((1-x^6)/(1+x))^2; g:= Reverse(f) div x; Coeffs := [Coefficient(g, i):i in [0..N]]; Coeffs; // Vincenzo Librandi, Mar 26 2026

Crossrefs

Cf. A393658, A393660.

Sequence in context: A126569 A162748 A374548 * A061815 A390481 A340361

Adjacent sequences: A393656 A393657 A393658 * A393660 A393661 A393662

Keyword

nonn

Author

Seiichi Manyama, Feb 24 2026

Status

approved