A100520 -id:A100520

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A100521

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

+10
2

1, 4, 72, 1200, 19600, 635040, 25613280, 82450368, 9275666400, 595703908800, 2048086772160, 23459903026560, 413676290035008, 4419618483280000, 3221901874311120000, 361282596839420256000, 2630246784565779288000, 9628029406360113091200, 1310481780310126504080000

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..675

FORMULA

a(n) = denominator( Sum_{k=0..2*n} (-1)^k/binomial(2*n, k)^2 ).

EXAMPLE

1, 7/4, 137/72, 2341/1200, 38629/19600, 1257937/635040, 50881679/25613280, 164078209/82450368, 18480100619/9275666400, 1187779852639/595703908800, ... = A100520/A100521

MATHEMATICA

Table[Denominator[Sum[(-1)^k/Binomial[2*n, k]^2, {k, 0, 2*n}]], {n, 0, 30}] (* G. C. Greubel, Jun 25 2022 *)

PROG

(Magma) [Denominator( (&+[(-1)^k/Binomial(2*n, k)^2: k in [0..2*n]]) ): n in [0..30]]; // G. C. Greubel, Jun 25 2022

(SageMath) [denominator(sum((-1)^k/binomial(2*n, k)^2 for k in (0..2*n))) for n in (0..30)] # G. C. Greubel, Jun 25 2022

(PARI) a(n) = denominator(sum(k=0, 2*n, (-1)^k/binomial(2*n, k)^2)); \\ Michel Marcus, Jun 25 2022

CROSSREFS

Cf. A100520.

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane, Nov 25 2004

EXTENSIONS

Definition corrected by Alexander Adamchuk, May 11 2007

STATUS

approved

A122184

Numerator of Sum_{k=0..2n} (-1)^k/C(2n,k)^3.

+10
0

1, 15, 1705, 47789, 1369377, 213162301, 43005554527, 14505995375, 23869750002797, 2384790127843063, 624724994927411, 24386251366041479501, 2042595777439018142725, 11191251831905709132993

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

p^k divides a((p^k+1)/2) for prime p>2 and integer k>0.

LINKS

Table of n, a(n) for n=0..13.

Eric Weisstein's World of Mathematics, Binomial Sums.

FORMULA

a(n) = Numerator[ Sum[ (-1)^k / Binomial[2n,k]^3, {k,0,2n} ] ].

MATHEMATICA

Table[ Numerator[ Sum[ (-1)^k / Binomial[2n, k]^3, {k, 0, 2n} ] ], {n, 0, 25} ]

CROSSREFS

Cf. A046825 = Numerator of Sum_{k=0..n} 1/C(n, k). Cf. A100516 = Numerator of Sum_{k=0..n} 1/C(n, k)^2. Cf. A100518 = Numerator of Sum_{k=0..n} 1/C(n, k)^3. Cf. A100520 = Numerator of Sum_{k=0..2n} (-1)^k/C(2n, k)^2.

KEYWORD

frac,nonn

AUTHOR

Alexander Adamchuk, May 10 2007

STATUS

approved

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