A184424 - OEIS

%I #17 Jun 09 2019 03:04:42

%S 1,6,252,15288,1089270,84963060,7023612960,604604070720,

%T 53620823521980,4865593245513000,449580815885401200,

%U 42156561463105471200,4001360292206427641400,383704407665664889683600

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

%H G. C. Greubel, <a href="/A184424/b184424.txt">Table of n, a(n) for n = 0..492</a>

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

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

%F 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

%e G.f.: A(x) = 1 + 6*x + 252*x^2 + 15288*x^3 + 1089270*x^4 +...

%e G.f. of A184423 equals A(x)^2:

%e A(x)^2 = 1 + 12*x + 540*x^2 + 33600*x^3 + 2425500*x^4 + 190702512*x^5 +...+ [(2n)!*(3n)!/n!^5]*x^n +...

%t a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/6, 1/3, 1, 108 x], {x, 0, n}]; (* _Michael Somos_, Sep 26 2011 *)

%t Table[3^n/(n!)^2 Product[(6k-4)(6k-5),{k,n}],{n,0,20}] (* _Harvey P. Dale_, May 10 2019 *)

%o (PARI) {a(n)=3^n*prod(k=1,n,(6*k-4)*(6*k-5))/n!^2}

%o (PARI) {a(n)=polcoeff(sum(m=0,n,(2*m)!*(3*m)!/m!^5*x^m+x*O(x^n))^(1/2),n)}

%Y Cf. A184423.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 13 2011