A393656 Expansion of (1/x) * Series_Reversion( x * ( Sum_{k=0..3} (-x)^k )^2 ).

1, 2, 5, 14, 44, 156, 611, 2550, 11001, 48224, 213325, 950820, 4271352, 19346480, 88335740, 406334918, 1881301815, 8759367346, 40982288984, 192561037552, 908216648372, 4298359134020, 20407008376965, 97164626491380, 463863430765254, 2219912272685016, 10647915018993791

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^4) / (1+x))^2 ).

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

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(2*n+k+1,k) * binomial(2*n+2,n-4*k).

Mathematica

CoefficientList[Normal@Series[InverseSeries@Series[x*((1-x^4)/(1+x))^2, {x, 0, 50}]/x, {x, 0, 27}], x] (* Vincenzo Librandi, Mar 24 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(2*n+k+1, k)*binomial(2*n+2, n-4*k))/(n+1);
(Magma) N := 25; R<x> := PowerSeriesRing(Rationals(), N+5); f:= x*((1-x^4)/(1+x))^2; g:= Reverse(f) div x; Coeffs := [Coefficient(g, i):i in [0..N]]; Coeffs; // Vincenzo Librandi, Mar 24 2026

Crossrefs

Cf. A063019, A393657.

Sequence in context: A095148 A368636 A060996 * A134378 A101226 A236043

Adjacent sequences: A393653 A393654 A393655 * A393657 A393658 A393659

Keyword

nonn

Author

Seiichi Manyama, Feb 24 2026

Status

approved