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A184424 a(n) = (3^n/n!^2) * Product_{k=1..n} (6k-4)*(6k-5). 6
1, 6, 252, 15288, 1089270, 84963060, 7023612960, 604604070720, 53620823521980, 4865593245513000, 449580815885401200, 42156561463105471200, 4001360292206427641400, 383704407665664889683600 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
LINKS
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 13 2011
STATUS
approved

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Last modified June 15 11:51 EDT 2024. Contains 373407 sequences. (Running on oeis4.)