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A276099 a(n) = (9*n)!*(5/2*n)!/((9*n/2)!*(5*n)!*(2*n)!). 5
1, 96, 24310, 7028736, 2149374150, 678057476096, 218191487357116, 71184392021606400, 23459604526110889542, 7791432263086689484800, 2603575153867220801823060, 874329826463740757819785216, 294822072977645830504963830300 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Let a > b be nonnegative integers. The ratio of factorials (2*a*n)!*(b*n)!/( (a*n)!*(2*b*n)!*((a - b)*n)! ) is known to be an integer for all integer n >= 0 (see, for example, Bober, Theorem 1.1). We have the companion result: Let a > b be nonnegative integers. Then the ratio of factorials ((2*a + 1)*n)!*((b + 1/2)*n)!/(((a + 1/2)*n)!*((2*b + 1)*n)!*((a - b)*n)!) is an integer for all integer n >= 0. This is the case a = 4, b = 2. Other cases include A091496 (a = 2, b = 0), A091527 (a = 1, b = 0), A262732 (a = 2, b = 1), A262733 (a = 3, b = 2) and A276098 (a = 3, b = 1).

REFERENCES

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

LINKS

Table of n, a(n) for n=0..12.

P. Bala, Some integer ratios of factorials

J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977v1 [math.NT], J. London Math. Soc., 79, Issue 2, (2009), 422-444.

FORMULA

a(n) = Sum_{k = 0..2*n} binomial(9*n, k)*binomial(7*n - k - 1, 2*n - k).

a(n) = Sum_{k = 0..n} binomial(14*n, 2*n - 2*k)*binomial(5*n + k - 1, k).

a(n) ~ 1/sqrt(4*Pi*n) * (3^18/5^5)^(n/2).

O.g.f. A(x) = Hypergeom([17/18, 13/18, 11/18, 7/18, 5/18, 1/18, 5/6, 1/6], [9/10, 7/10, 3/10, 1/10, 3/4, 1/4, 1/2], (3^18/5^5)*x^2) + 96*x*Hypergeom([13/9, 11/9, 10/9, 8/9, 7/9, 5/9, 4/3, 2/3], [7/5, 6/5, 4/5, 3/5, 5/4, 3/4, 3/2], (3^18/5^5)*x^2).

a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^9/(1 - x)^5.

It follows that the o.g.f. A(x) for this sequence is the diagonal of the bivariate rational generating function 1/2*( 1/(1 - t*H(sqrt(x))) + 1/(1 - t*H(-sqrt(x))) ) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197.

Let F(x) = 1/x*Series_Reversion( x*sqrt((1 - x)^5/(1 + x)^9) ) and put G(x) = 1 + x*d/dx(Log(F(x)). Then A(x^2) = (G(x) + G(-x))/2.

MAPLE

seq(simplify((9*n)!*(5/2*n)!/((9*n/2)!*(5*n)!*(2*n)!)), n = 0..20);

CROSSREFS

Cf. A091496, A091527, A262732, A262733, A276098.

Sequence in context: A295597 A203489 A208794 * A233161 A269090 A232522

Adjacent sequences:  A276096 A276097 A276098 * A276100 A276101 A276102

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Aug 22 2016

STATUS

approved

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Last modified September 17 11:15 EDT 2019. Contains 327129 sequences. (Running on oeis4.)