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A001421 (6*n)!/((n!)^3*(3*n)!). 5
1, 120, 83160, 81681600, 93699005400, 117386113965120, 155667030019300800, 214804163196079142400, 305240072216678400087000, 443655767845074392936328000, 656486312795713480715743268160 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Self-convolution of A092870, where A092870(n) = (12^n/n!^2) * Product_{k=0..n-1} (12k+1)*(12k+5). - Paul D. Hanna, Jan 25 2011

REFERENCES

M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998. (See Eq. 31.)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..75

R. S. Maier, Nonlinear differential equations satisfied by certain classical modular forms, p. 34 equation (7.29b)

FORMULA

o.g.f.: Hypergeometric2F1(5/12, 1/12; 1; 1728x)^2. - Jacob Lewis (jacobml(AT)uw.edu), Jul 28 2009

a(n) = C(2n,n) * (12^n/n!^2) * Product_{k=0..n-1} (6k+1)*(6k+5). - Paul D. Hanna, Jan 25 2011

G.f.: A(x) = 1 + 120*x + 83160*x^2 + 81681600*x^3 +...  - Paul D. Hanna, Jan 25 2011

A(x)^(1/2) = 1 + 60*x + 39780*x^2 + 38454000*x^3 +...+ A092870(n)*x^n +... - Paul D. Hanna, Jan 25 2011

G.f.: F(1/6, 1/2, 5/6; 1, 1; 1728*x), a hypergeometric series. - Michael Somos, Feb 28 2011

0 = y^3*z^3 - 360*y^4*z^2 + 43200*y^5*z - 1728000*y^6 - 16632*x*y^2*z^3 + 7691328*x*y^3*z^2 - 1738520064*x*y^4*z + 176027074560*x*y^5 + 92207808*x^2*y*z^3 - 69176553984*x^2*y^2*z^2 + 23624298528768*x^2*y^3*z - 2853152143441920*x^2*y^4 - 170400029184*x^3*z^3 + 224945232150528*x^3*y*z^2 - 92759146352345088*x^3*y^2*z + 11686511179538104320*x^3*y^3 where x = a(n), y = a(n+1), z = a(n+2) for all n in z. - Michael Somos, Sep 21 2014

MAPLE

f := n->(6*n)!/( (n!)^3*(3*n)!);

MATHEMATICA

Factorial[6 n]/(Factorial[3n] Factorial[n]^3) (* Jacob Lewis (jacobml(AT)uw.edu), Jul 28 2009 *)

a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/6, 1/2, 5/6}, {1, 1}, 1728 x], {x, 0, n}] (* Michael Somos, Jul 11 2011 *)

PROG

(PARI) {a(n)=(2*n)!/n!^2*(12^n/n!^2)*prod(k=0, n-1, (6*k+1)*(6*k+5))} \\ Paul D. Hanna, Jan 25 2011

(MAGMA) [Factorial(6*n)/(Factorial(n)^3*Factorial(3*n)): n in [0..15]]; // Vincenzo Librandi, Oct 26 2011

CROSSREFS

Cf. A092870; variants: A184423, A008977, A184892, A184896, A184898. - Paul D. Hanna, Jan 25 2011

Sequence in context: A074653 A065961 A058528 * A107446 A184887 A159735

Adjacent sequences:  A001418 A001419 A001420 * A001422 A001423 A001424

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, KUPK78A(AT)prodigy.com (Glenn K Painter)

STATUS

approved

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Last modified August 30 04:08 EDT 2015. Contains 261200 sequences.