A100517 Denominator of Sum_{k=0..n} 1/binomial(n,k)^2.

1, 1, 4, 9, 72, 10, 3600, 1575, 2800, 1764, 14112, 13475, 34927200, 2316600, 192192, 4459455, 4994589600, 262061800, 735869534400, 17476901442, 422721728, 353723760, 31127690880, 10150725585, 59637542956992, 2205530434800, 155748568976000, 50956005028500

Offset

0,3

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

Formula

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

Example

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

Mathematica

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

Prog

(Magma) [Denominator( (&+[1/Binomial[n, k]^2: k in [0..n]]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022
(SageMath) [denominator(sum(1/binomial(n, k)^2 for k in (0..n))) for n in (0..40)] # G. C. Greubel, Jun 24 2022
(PARI) a(n) = denominator(sum(k=0, n, 1/binomial(n, k)^2)); \\ Michel Marcus, Jun 24 2022

Crossrefs

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

Sequence in context: A122956 A041777 A367276 * A327579 A041597 A041030

Adjacent sequences: A100514 A100515 A100516 * A100518 A100519 A100520

Keyword

nonn,frac

Author

N. J. A. Sloane, Nov 25 2004

Status

approved