A388538 Expansion of (1/x) * Series_Reversion( x / ((1+x)^6 + x^2) ).

1, 6, 52, 524, 5759, 66952, 809613, 10079502, 128332436, 1663258710, 21870751170, 291054084778, 3912635825360, 53053042667956, 724747188351807, 9965196179924218, 137806606441318292, 1915394528899038232, 26743280989671215987, 374920779552993761604, 5275478128010959296365

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..851

Formula

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

a(n) = (1/(n+1)) * [x^n] ((1+x)^6 + x^2)^(n+1).

Mathematica

Table[(1/(n+1)) Coefficient[((1+x)^6+x^2)^(n+1), x, n], {n, 0, 20}] (* Vincenzo Librandi, Sep 27 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^6+x^2))/x)
(PARI) a(n) = sum(k=0, n\2, binomial(n+1, k)*binomial(6*n-6*k+6, n-2*k))/(n+1);
(Magma) R<x> := PolynomialRing(Rationals()); [ (1/(n+1))*Coefficient(((1+x)^6 + x^2)^(n+1), n) : n in [0..20] ]; // Vincenzo Librandi, Sep 27 2025

Crossrefs

Cf. A065065, A071356, A369213, A388532.

Cf. A378691, A388537.

Sequence in context: A084773 A357154 A390384 * A365194 A365755 A232821

Adjacent sequences: A388535 A388536 A388537 * A388539 A388540 A388541

Keyword

nonn

Author

Seiichi Manyama, Sep 18 2025

Status

approved