A100516 Numerator of Sum_{k=0..n} 1/binomial(n,k)^2.

1, 2, 9, 20, 155, 21, 7441, 3224, 5697, 3575, 28523, 27183, 70357417, 4661447, 386395, 8959408, 10028928779, 525966759, 1476346738309, 35051863075, 847581175, 709068173, 62385202783, 20340152122, 119483756745025, 4418168441921, 311960929172031

Offset: 0

Author

N. J. A. Sloane, Nov 25 2004

Keywords

• nonn
• frac

Examples

			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
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A046825, A100517, A100518.

Programs

Magma

[Numerator( (&+[1/Binomial(n,k)^2: k in [0..n]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022

Mathematica

Table[3*(n+1)^2/((n+2)*(2*n+3)*CatalanNumber[n+1])*Sum[((k+ 1)/k)*CatalanNumber[k], {k,n+1}], {n,0,40}]//Numerator (* G. C. Greubel, Jun 24 2022 *)

PARI

a(n) = numerator(sum(k=0, n, 1/binomial(n,k)^2)); \\ Michel Marcus, Jun 24 2022

SageMath

[numerator(sum(1/binomial(n,k)^2 for k in (0..n))) for n in (0..40)] # G. C. Greubel, Jun 24 2022

Formula

a(n) = numerator( 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