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A184423 a(n) = (2*n)!*(3*n)!/n!^5. 6
1, 12, 540, 33600, 2425500, 190702512, 15849497664, 1369618398720, 121821136479900, 11079206239530000, 1025579963180813040, 96310511463483233280, 9152842704012278107200, 878622906816654279840000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Denoted by h_3[n] by T. Piezas III. He also gives formulas for 1/Pi such as 1/Pi = 2 * Sum_{n>=0} a(n) * (-1)^n * (51*n + 7) / (12^3)^(n + 1/2). - Michael Somos, May 31 2012

Diagonal of the rational function R(x,y,z,w) = 1/(1-(w*y+w*z+x+y+z)). - Gheorghe Coserea, Jul 15 2016

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 0..200

T. Piezas III, 0013: Article 3 (Pi Formulas and the Monster Group).

FORMULA

Self-convolution of A184424:

a(n) = Sum_{k=0..n} A184424(k)*A184424(n-k) where A184424(n) = (3^n/n!^2)*Product_{k=1..n} (6*k-4)*(6*k-5).

a(n) = 6 * (2*n - 1) * (3*n - 1) * (3*n - 2) / n^3 * a(n-1) if n>0. - Michael Somos, May 31 2012

0 = (x^2-108*x^3)*y''' + (3*x-486*x^2)*y''+ (1-348*x)*y' - 12*y, where y is g.f. - Gheorghe Coserea, Jul 15 2016

a(n) ~ 3^(1/2)/(2*Pi^(3/2)) * n^(-3/2) * 108^n. - Ilya Gutkovskiy, Jul 15 2016

EXAMPLE

G.f.: A(x) = 1 + 12*x + 540*x^2 + 33600*x^3 + 2425500*x^4 +...

G.f. of A184424 equals A(x)^(1/2):

A(x)^(1/2) = 1 + 6*x + 252*x^2 + 15288*x^3 + 1089270*x^4 + 84963060*x^5 +...+ [(3^n/n!^2)*Product_{k=1..n} (6*k-4)*(6*k-5)]*x^n +...

PROG

(PARI) {a(n)=(3*n)!*(2*n)!/n!^5}

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

CROSSREFS

Cf. A184424.

Related to diagonal of rational functions: A268545-A268555.

Sequence in context: A004801 A202079 A067733 * A064344 A163046 A193381

Adjacent sequences:  A184420 A184421 A184422 * A184424 A184425 A184426

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 13 2011

STATUS

approved

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Last modified December 2 19:13 EST 2016. Contains 278683 sequences.