A388537 Coefficient of x^n in the expansion of ( (1+x)^6 + x^2 )^n.

1, 6, 68, 852, 11244, 152806, 2116382, 29703372, 420989052, 6011825496, 86364467658, 1246724231220, 18069698843958, 262785225921120, 3832704661139884, 56039343268125352, 821155990644795468, 12055627565174426168, 177292905570172198832, 2611266841812710089200

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(6*n-6*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)^6 + x^2) ). See A388538.

Mathematica

Table[SeriesCoefficient[Series[((1+t)^6+t^2)^n, {t, 0, n}], n], {n, 0, 20}] (* Vincenzo Librandi, Sep 27 2025 *)

Prog

(PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(6*n-6*k, n-2*k));
(Magma) R<t> := PolynomialRing(Integers()); seq := [ MonomialCoefficient(((1+t)^6 + t^2)^n, t^n) : n in [0..20] ]; seq; // Vincenzo Librandi, Sep 27 2025

Crossrefs

Cf. A006139, A370286, A388530, A388531.

Cf. A388536, A388538.

Sequence in context: A186669 A258134 A256238 * A359714 A370938 A349557

Adjacent sequences: A388534 A388535 A388536 * A388538 A388539 A388540

Keyword

nonn

Author

Seiichi Manyama, Sep 18 2025

Status

approved