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A061163
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a(n) = (10n)!*n!/((5n)!*(4n)!*(2n)!).
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8
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1, 630, 1385670, 3528923580, 9540949030470, 26651569523959380, 75998432812419471900, 219813190240007470094520, 642409325786050322446410310, 1892390644737640220059489996260
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OFFSET
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0,2
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COMMENTS
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According to page 781 of the cited reference the generating function F(x) for a(n) is algebraic but not obviously so and the minimal polynomial satisfied by F(x) is quite large.
This sequence is the particular case a = 5, b = 1 of the following result (see Bober, Theorem 1.2): let a, b be nonnegative integers with a > b and GCD(a,b) = 1. Then (2*a*n)!*(b*n)!/((a*n)!*(2*b*n)!*((a-b)*n)!) is an integer for all integer n >= 0. Other cases include A061162 (a = 3, b = 1), A211419 (a = 3, b = 2) and A211420(a = 4, b = 1) and A211421 (a = 4, b = 3). The o.g.f. sum {n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas). - Peter Bala, Apr 10 2012
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REFERENCES
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M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.
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LINKS
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Table of n, a(n) for n=0..9.
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977 [math.NT], 2007; Jour. of the London Math. Soc., Vol. 79, Issue 2, 422-444.
F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.
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FORMULA
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n*(4*n-3)*(2*n-1)*(4*n-1)*a(n) -10*(10*n-9)*(10*n-7)*(10*n-3)*(10*n-1)*a(n-1)=0. - R. J. Mathar, Oct 26 2014
O.g.f. is a generalized hypergeometric function 4F3([1/10, 3/10, 7/10, 9/10], [1/4, 1/2, 3/4], 5^5*z). - Karol A. Penson, Apr 13 2022
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MAPLE
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A061123 := n->(10*n)!*n!/((5*n)!*(4*n)!*(2*n)!);
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MATHEMATICA
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Table[(10n)! n!/((5n)!(4n)!(2n)!), {n, 0, 10}] (* Harvey P. Dale, Oct 24 2022 *)
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CROSSREFS
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Cf. A061163, A061164. A211419, A211420, A211421.
Sequence in context: A185849 A058832 A225390 * A045168 A270802 A119504
Adjacent sequences: A061160 A061161 A061162 * A061164 A061165 A061166
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KEYWORD
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easy,nonn
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AUTHOR
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Richard Stanley, Apr 17 2001
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STATUS
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approved
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