This site is supported by donations to The OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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