A370099 a(n) = Sum_{k=0..n} binomial(2*n,k) * binomial(3*n-k-1,n-k).

1, 4, 32, 292, 2816, 28004, 284000, 2919620, 30316544, 317222212, 3339504032, 35329425124, 375282559232, 4000059761572, 42760427177696, 458259268924292, 4921911787962368, 52965710906750084, 570951048018417440, 6164049197776406180, 66639047280436354816

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

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

a(n) = 2 * A103885(n) for n >= 1. - Peter Bala, Sep 16 2024

From Seiichi Manyama, Aug 09 2025: (Start)

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

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

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

Prog

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

Crossrefs

Cf. A370098, A370102.

Cf. A032349, A103885.

Sequence in context: A295538 A256183 A000766 * A392524 A267982 A231258

Adjacent sequences: A370096 A370097 A370098 * A370100 A370101 A370102

Keyword

nonn,easy

Author

Seiichi Manyama, Feb 10 2024

Status

approved