A386766 a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(n+k-1,k).

1, 7, 99, 1618, 28051, 502182, 9174174, 169955268, 3180814851, 59997194782, 1138669104874, 21718428172668, 415955669988526, 7994062687411132, 154087546950639324, 2977629771383522568, 57667991491752308451, 1119034767346120619982, 21752068061568290996274

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

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

a(n) = [x^n] ( (1+2*x)^2/(1-3*x) )^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-3*x) / (1+2*x)^2 ). See A386772.

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

D-finite with recurrence 9*n*a(n) +6*(-18*n-5)*a(n-1) +80*(-17*n+37)*a(n-2) +800*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Aug 03 2025

Prog

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

Crossrefs

Cf. A386767, A386768.

Cf. A386772.

Sequence in context: A238472 A041087 A041084 * A115066 A385441 A340887

Adjacent sequences: A386763 A386764 A386765 * A386767 A386768 A386769

Keyword

nonn

Author

Seiichi Manyama, Aug 02 2025

Status

approved