A370105 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k-1,k) * binomial(5*n-k-1,n-k).

1, 3, 23, 192, 1687, 15253, 140504, 1311292, 12357015, 117318162, 1120436273, 10752242592, 103596191608, 1001494496863, 9709576926716, 94369011385192, 919175964169623, 8970063281146830, 87685232945278010, 858446087522807784, 8415669293820893937

Offset

0,2

Links

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

Formula

a(n) = [x^n] 1/( (1+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)*(1-x)^4 ). See A365752.

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

Prog

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

Crossrefs

Cf. A348410, A352373.

Cf. A365752.

Sequence in context: A060880 A331718 A093155 * A372506 A241886 A096649

Adjacent sequences: A370102 A370103 A370104 * A370106 A370107 A370108

Keyword

nonn

Author

Seiichi Manyama, Feb 10 2024

Status

approved