A370103 a(n) = Sum_{k=0..n} (-1)^k * binomial(2*n+k-1,k) * binomial(4*n-k-1,n-k).

1, 1, 7, 28, 151, 751, 3976, 20924, 112023, 602182, 3260257, 17724928, 96766072, 529977917, 2910984412, 16027963528, 88440034711, 488918693466, 2707393587802, 15014647096172, 83380131228401, 463593653171495, 2580426581343200, 14377474236172320

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = [x^n] 1/( (1+x)^2 * (1-x)^3 )^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)^3 ). See A365854.

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

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

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(2*n+k-1, k)*binomial(4*n-k-1, n-k));
(PARI) a(n, s=2, t=3, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial(u*n, n-s*k));
(PARI) a(n, s=2, t=2, u=1) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((u+1)*n-s*k-1, n-s*k));

Crossrefs

Cf. A351857, A370104.

Cf. A352373, A370098, A370101.

Cf. A001700, A348410.

Cf. A365854.

Sequence in context: A025030 A001554 A370243 * A359723 A026664 A304520

Adjacent sequences: A370100 A370101 A370102 * A370104 A370105 A370106

Keyword

nonn

Author

Seiichi Manyama, Feb 10 2024

Status

approved