A366400 G.f. A(x) satisfies A(x) = (1 + x * A(x)^(5/2)) / (1 - x).

1, 2, 7, 32, 167, 942, 5593, 34438, 217888, 1407938, 9252168, 61641846, 415412036, 2826736736, 19395080061, 134034296976, 932110471089, 6518146460274, 45805553781349, 323313555424924, 2291130483593189, 16294149468133930, 116259325138469680

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

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

From Seiichi Manyama, Dec 12 2024: (Start)

G.f. A(x) satisfies:

(1) A(x) = ( 1 + x*A(x)^2/(1 + x*A(x)) )^2.

(2) A(x) = 1/( 1 - x*A(x)^(3/2)/(1 + x*A(x)) )^2.

(3) A(x) = 1 + x * A(x) * (1 + A(x)^(3/2)).

(4) A(x) = B(x)^2 where B(x) is the g.f. of A219537.

If g.f. satisfies A(x) = ( 1 + x*A(x)^(t/r) * (1 + x*A(x)^(u/r))^s )^r, then a(n) = r * Sum_{k=0..n} binomial(t*k+u*(n-k)+r,k) * binomial(s*k,n-k)/(t*k+u*(n-k)+r). (End)

G.f.: Sum_{k>=0} binomial(5*k/2, k)*x^k/((3*k/2 + 1)*(1 - x)^(5*k/2 + 1)). - Miles Wilson, Feb 02 2025

Prog

(PARI) a(n) = sum(k=0, n, binomial(n+3*k/2, n-k)*binomial(5*k/2, k)/(3*k/2+1));
(PARI) a(n, r=2, s=-1, t=4, u=2) = r*sum(k=0, n, binomial(t*k+u*(n-k)+r, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+r)); \\ Seiichi Manyama, Dec 12 2024

Crossrefs

Cf. A349311, A366401, A366402, A366403, A366404, A366405, A366406, A366407.

Cf. A262441, A370474.

Cf. A219537, A378890.

Sequence in context: A363562 A263532 A108524 * A226269 A006781 A330075

Adjacent sequences: A366397 A366398 A366399 * A366401 A366402 A366403

Keyword

nonn

Author

Seiichi Manyama, Oct 09 2023

Status

approved