A388530 Coefficient of x^n in the expansion of ( (1+x)^4 + x^2 )^n.

1, 4, 30, 244, 2090, 18424, 165486, 1506068, 13840458, 128146600, 1193557780, 11170608420, 104965939754, 989655783016, 9357723121854, 88702574743084, 842644874094986, 8020146961241832, 76463764316083884, 730105973920973688, 6980817414644155740

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Formula

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^4 + x^2) ). See A369213.

Mathematica

Table[SeriesCoefficient[Series[((1+t)^4+t^2)^n, {t, 0, n}], n], {n, 0, 20}] (* Vincenzo Librandi, Sep 26 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(4*n-4*k, n-2*k));
(Magma) R<t> := PolynomialRing(Integers()); seq := [ MonomialCoefficient(((1+t)^4 + t^2)^n, t^n) : n in [0..20] ]; seq; // Vincenzo Librandi, Sep 26 2025

Crossrefs

Cf. A006139, A370286, A388531, A388537.

Cf. A359643, A369213.

Sequence in context: A213102 A052604 A391492 * A390338 A357204 A038225

Adjacent sequences: A388527 A388528 A388529 * A388531 A388532 A388533

Keyword

nonn

Author

Seiichi Manyama, Sep 18 2025

Status

approved