A395779 Expansion of (Sum_{k>=0} binomial(3*k+1,k) * x^k)^3.

1, 12, 111, 928, 7356, 56436, 423579, 3129552, 22851201, 165333648, 1187526564, 8478925056, 60241049848, 426221826564, 3004932019443, 21120318379488, 148048546150899, 1035347613355212, 7225423763357031, 50330772078599808, 350009710750032624, 2430378171200375472

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = Sum_{i,j,k >= 0 and i+j+k=n} binomial(3*i+1,i) * binomial(3*j+1,j) * binomial(3*k+1,k).

G.f.: B(x)^3 where B(x) is the g.f. of A045721.

a(n) = Sum_{k=0..n} (k+1) * 2^k * binomial(3*n+5,n-k).

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

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (3*k+12) * 2^k * binomial(k+2,2) * binomial(3*n+5,n-1-k).

a(0) = 1; a(n) = (1/n) * Sum_{k=0..n-1} (2*k+12) * 3^k * binomial(k+2,2) * binomial(3*n+2-k,n-1-k).

Prog

(PARI) a(n) = sum(k=0, n, (k+1)*2^k*binomial(3*n+5, n-k));

Crossrefs

Cf. A045721, A062236, A395780.

Cf. A378483, A395783.

Sequence in context: A075231 A085773 A066042 * A153597 A036733 A253091

Adjacent sequences: A395776 A395777 A395778 * A395780 A395781 A395782

Keyword

nonn,easy

Author

Seiichi Manyama, May 05 2026

Status

approved