A393657 Expansion of (1/x) * Series_Reversion( x * ( Sum_{k=0..3} (-x)^k )^3 ).

1, 3, 12, 55, 276, 1482, 8382, 49335, 299367, 1859628, 11764272, 75506418, 490337696, 3215349768, 21258396852, 141547280455, 948302613738, 6387760541085, 43235909268972, 293910775409340, 2005742871137616, 13736126242754986, 94372306795872780, 650273379357528810

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))^3 ).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(3*n+k+2,k) * binomial(4*n-2*k+2,n-2*k).

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} binomial(3*n+k+2,k) * binomial(3*n+3,n-4*k).

Mathematica

CoefficientList[Normal@Series[InverseSeries@Series[x*((1-x^4)/(1+x))^3, {x, 0, 50}]/x, {x, 0, 27}], x] (* Vincenzo Librandi, Mar 25 2026 *)

Prog

(PARI) a(n) = sum(k=0, n\4, binomial(3*n+k+2, k)*binomial(3*n+3, n-4*k))/(n+1);
(Magma) N := 25; R<x> := PowerSeriesRing(Rationals(), N+5); f:= x*((1-x^4)/(1+x))^3; g:= Reverse(f) div x; Coeffs := [Coefficient(g, i):i in [0..N]]; Coeffs; // Vincenzo Librandi, Mar 25 2026

Crossrefs

Cf. A063019, A393656.

Sequence in context: A120920 A179487 A350265 * A263533 A064314 A362085

Adjacent sequences: A393654 A393655 A393656 * A393658 A393659 A393660

Keyword

nonn

Author

Seiichi Manyama, Feb 24 2026

Status

approved