A184424 a(n) = (3^n/n!^2) * Product_{k=1..n} (6k-4)*(6k-5).

1, 6, 252, 15288, 1089270, 84963060, 7023612960, 604604070720, 53620823521980, 4865593245513000, 449580815885401200, 42156561463105471200, 4001360292206427641400, 383704407665664889683600

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..492

Formula

Self-convolution equals A184423, where A184423(n) = (2n)!*(3n)!/n!^5:

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

a(n) ~ 2^(2*n + 1/3) * 3^(3*n - 1/2) * sqrt(Pi) / (Gamma(1/3)^3 * n^(3/2)). - Vaclav Kotesovec, Jun 09 2019

Example

G.f.: A(x) = 1 + 6*x + 252*x^2 + 15288*x^3 + 1089270*x^4 +...
G.f. of A184423 equals A(x)^2:
A(x)^2 = 1 + 12*x + 540*x^2 + 33600*x^3 + 2425500*x^4 + 190702512*x^5 +...+ [(2n)!*(3n)!/n!^5]*x^n +...

Mathematica

a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/6, 1/3, 1, 108 x], {x, 0, n}]; (* Michael Somos, Sep 26 2011 *)
Table[3^n/(n!)^2 Product[(6k-4)(6k-5), {k, n}], {n, 0, 20}] (* Harvey P. Dale, May 10 2019 *)

Prog

(PARI) {a(n)=3^n*prod(k=1, n, (6*k-4)*(6*k-5))/n!^2}
(PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!*(3*m)!/m!^5*x^m+x*O(x^n))^(1/2), n)}

Crossrefs

Cf. A184423.

Sequence in context: A335158 A221822 A361186 * A324478 A230881 A332563

Adjacent sequences: A184421 A184422 A184423 * A184425 A184426 A184427

Keyword

nonn

Author

Paul D. Hanna, Jan 13 2011

Status

approved