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A178529
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Self-convolution square-root of A008977, where A008977(n) = (4n)!/(n!)^4.
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6
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1, 12, 1188, 170544, 28779300, 5318414640, 1041818334480, 212530940233920, 44671347000417060, 9607097095645249200, 2103954263946309574800, 467599488149125265169600, 105196895958882375628016400
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OFFSET
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0,2
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COMMENTS
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In Narumiya and Shiga on bottom of page 157 the g.f. is given as an integral. On page 158 the square of the g.f. is given as a hypergeometric function. - Michael Somos, Aug 12 2014
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REFERENCES
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N. Narumiya and H. Shiga, "The mirror map for a family of K3 surfaces induced from the simplest 3-dimensional reflexive polytope", Proceedings on Moonshine and related topics (Montréal, QC, 1999), 139-161, CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001. MR1877764 (2002m:14030)
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LINKS
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FORMULA
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a(n) = 4^n/(n!)^2 * Product_{k=0..n-1} (8*k+1)*(8*k+3).
a(n) = 2^(8*n) * GAMMA(n+1/8) * GAMMA(n+3/8) /(GAMMA(1/8)*GAMMA(3/8) *GAMMA(n+1)^2). - Vaclav Kotesovec, Mar 07 2014
a(n) ~ GAMMA(5/8)*GAMMA(7/8) * 2^(8*n-3/2) / (Pi^2 * n^(3/2)). - Vaclav Kotesovec, Mar 07 2014
G.f.: F( 1/8, 3/8, 1; x) = 1 / B(3/8, 5/8) * integral_0^1 (u^5 * (1-u)^3 * (1-x*u))^(-1/8) du. - Michael Somos, Aug 12 2014
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EXAMPLE
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G.f.: A(x) = 1 + 12*x + 1188*x^2 + 170544*x^3 + 28779300*x^4 +...
A(x)^2 = 1 + 24*x + 2520*x^2 + 369600*x^3 +...+ (4n)!/(n!)^4*x^n +...
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MATHEMATICA
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Table[4^n/(n!)^2*Product[(8*k + 1)*(8*k + 3), {k, 0, n - 1}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 07 2014 *)
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/8, 3/8, 1, 256 x], {x, 0, n}]; (* Michael Somos, Aug 12 2014 *)
a[ n_] := 256^n / n!^2 Pochhammer[ 1/8, n] Pochhammer[ 3/8, n]; (* Michael Somos, Aug 12 2014 *)
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PROG
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(PARI) {a(n)=4^n*prod(k=0, n-1, (8*k+1)*(8*k+3))/(n!)^2}
(PARI) {a(n)=polcoeff(sqrt(sum(k=0, n, (4*k)!/(k!)^4*x^k)+x*O(x^n)), n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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