A388728 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n+4*k,k).

1, 8, 112, 1778, 29776, 513833, 9039286, 161164599, 2902000592, 52652740496, 961069474297, 17628176117318, 324649112983798, 5999256527934392, 111183502998164219, 2065716744515378213, 38463565347521894032, 717567269332838675232, 13409586559291172964976

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..750

Formula

a(n) = [x^n] ((1+x)^3 * (x+(1+x)^4))^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^3 * (x+(1+x)^4)) ).

Mathematica

Table[Sum[Binomial[ n, k]*Binomial[3*n+4*k, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 22 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(3*n+4*k, k));
(Magma) [&+[Binomial(n, k)*Binomial(3*n+4*k, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 22 2025

Crossrefs

Cf. A388535, A388727, A388729.

Cf. A156886. A388726.

Cf. A378685.

Sequence in context: A067414 A265665 A386918 * A302104 A219184 A034689

Adjacent sequences: A388725 A388726 A388727 * A388729 A388730 A388731

Keyword

nonn

Author

Seiichi Manyama, Sep 20 2025

Status

approved