A100517 - OEIS

%I #14 Jan 17 2025 02:12:40

%S 1,1,4,9,72,10,3600,1575,2800,1764,14112,13475,34927200,2316600,

%T 192192,4459455,4994589600,262061800,735869534400,17476901442,

%U 422721728,353723760,31127690880,10150725585,59637542956992,2205530434800,155748568976000,50956005028500

%N Denominator of Sum_{k=0..n} 1/binomial(n,k)^2.

%D H. W. Gould, Combinatorial Identities, Morgantown, 1972, p. 50, formula (5.2).

%H G. C. Greubel, <a href="/A100517/b100517.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = denominator( 3*(n+1)^2/((n+2)*(2*n+3)*Catalan(n+1)) * Sum_{k=1..n+1} binomial(2*k, k)/k ). - _G. C. Greubel_, Jun 24 2022

%e 1, 2, 9/4, 20/9, 155/72, 21/10, 7441/3600, 3224/1575, 5697/2800, 3575/1764, 28523/14112, 27183/13475, 70357417/34927200, 4661447/2316600, ... = A100516/A100517

%t Table[Sum[1/Binomial[n,k]^2,{k,0,n}],{n,0,30}]//Denominator (* _Harvey P. Dale_, Apr 01 2019 *)

%o (Magma) [Denominator( (&+[1/Binomial(n,k)^2: k in [0..n]]) ): n in [0..40]]; // _G. C. Greubel_, Jun 24 2022

%o (SageMath) [denominator(sum(1/binomial(n,k)^2 for k in (0..n))) for n in (0..40)] # _G. C. Greubel_, Jun 24 2022

%o (PARI) a(n) = denominator(sum(k=0, n, 1/binomial(n,k)^2)); \\ _Michel Marcus_, Jun 24 2022

%Y Cf. A000108, A046825, A046826, A100516, A100518, A100519.

%K nonn,frac

%O 0,3

%A _N. J. A. Sloane_, Nov 25 2004