A100516 -id:A100516 - cp's OEIS frontend

A100518 Numerator of Sum_{k=0..n} 1/binomial(n,k)^3.

1, 2, 17, 56, 1759, 1009, 86831, 2322304, 85922, 1144667, 16019198113, 123357293, 21312406359367, 17061774340031, 27741170437991, 182851619022848, 167169857863289, 9857517443932187, 8844183281912559671, 197147246106875452361, 681198614358931646209

Offset: 0

N. J. A. Sloane, Nov 25 2004

			1, 2, 17/8, 56/27, 1759/864, 1009/500, 86831/43200, 2322304/1157625, 85922/42875, 1144667/571536, 16019198113/8001504000, 123357293/61631955, ... = A100518/A100519.

G. C. Greubel, Table of n, a(n) for n = 0..770

Cf. A046825, A046826, A100516, A100517, A100519.

[Numerator( (&+[1/Binomial(n,k)^3: k in [0..n]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022

Numerator[Table[Sum[1/Binomial[n,k]^3,{k,0,n}],{n,0,20}]] (* Harvey P. Dale, Sep 28 2012 *)

a(n) = numerator(sum(k=0, n, 1/binomial(n,k)^3)); \\ Michel Marcus, Jun 24 2022

[numerator(sum(1/binomial(n,k)^3 for k in (0..n))) for n in (0..40)] # G. C. Greubel, Jun 24 2022

a(n) = numerator( Sum_{k=0..n} 1/binomial(n,k)^3 ).

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

Original entry on oeis.org

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

N. J. A. Sloane, Nov 25 2004

			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

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

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

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

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

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

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

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

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

A100519 Denominator of Sum_{k=0..n} 1/binomial(n,k)^3.

Original entry on oeis.org

1, 1, 8, 27, 864, 500, 43200, 1157625, 42875, 571536, 8001504000, 61631955, 10650001824000, 8526987612000, 13865513485824, 91398648466125, 83564478597600, 4927753743913000, 4421332282230864000, 98559233902419862572, 340556687709473664000

Offset: 0

N. J. A. Sloane, Nov 25 2004

			1, 2, 17/8, 56/27, 1759/864, 1009/500, 86831/43200, 2322304/1157625, 85922/42875, 1144667/571536, 16019198113/8001504000, 123357293/61631955, ... = A100518/A100519.

G. C. Greubel, Table of n, a(n) for n = 0..750

Cf. A046825, A100516, A100517, A100518.

[Denominator( (&+[1/Binomial(n,k)^3: k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, Jun 24 2022

Table[Denominator[Sum[1/Binomial[n,k]^3, {k,0,n}]], {n,0,30}] (* G. C. Greubel, Jun 24 2022 *)

a(n) = denominator(sum(k=0, n, 1/binomial(n,k)^3)); \\ Michel Marcus, Jun 25 2022

[denominator(sum(1/binomial(n,k)^3 for k in (0..n))) for n in (0..30)] # G. C. Greubel, Jun 24 2022

a(n) = denominator( Sum_{k=0..n} 1/binomial(n,k)^3 ).

A100520 Numerator of Sum_{k=0..2n} (-1)^k/binomial(2n, k)^2.

Original entry on oeis.org

1, 7, 137, 2341, 38629, 1257937, 50881679, 164078209, 18480100619, 1187779852639, 4086043585673, 46823724627623, 825926870076593, 8826243587390221, 6435629123661395137, 721766119107018403553, 5255377541226932317019, 19239461977895120106181, 2618947765106118753941303

Offset: 0

N. J. A. Sloane, Nov 25 2004

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

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

Cf. A100516, A100521.

[Numerator( (&+[(-1)^k/Binomial(2*n,k)^2: k in [0..2*n]]) ): n in [0..30]]; // G. C. Greubel, Jun 24 2022

Table[Numerator[Sum[(-1)^k/Binomial[2*n,k]^2, {k,0,2*n}]], {n,0,30}] (* G. C. Greubel, Jun 24 2022 *)

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

[numerator(sum((-1)^k/binomial(2*n,k)^2 for k in (0..2*n))) for n in (0..30)] # G. C. Greubel, Jun 24 2022

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

Definition corrected by Alexander Adamchuk, May 11 2007

A279055 Self-convolution of squares of factorial numbers (A001044).

Original entry on oeis.org

1, 2, 9, 80, 1240, 30240, 1071504, 51996672, 3307723776, 266872320000, 26615381760000, 3214252921651200, 462189467175321600, 78024380924038348800, 15279632043682406400000, 3435553774431004262400000, 879010223384483132866560000, 253916900613208108255150080000

Offset: 0

Arman Maesumi, Dec 04 2016

nonn

a(n) = (n!)^2 * Sum_{i=0..n} (binomial(n,i)^(-2)).
Consider a triangle ABC with area p. Let points X, Y, Z be randomly and uniformly chosen on sides BC, CA, BA. Let r = area of XYZ. Then the average or expected value of (r/p)^n = a(n)/(n!^2 * (n+1)^3).
a(n) = (3*(n+1)^4 *(n!)^4 /(2n+3)!) * Sum_{i=1..n+1} ((1/i)* binomial(2i, i)), see Sprugnoli Formula 5.2 as noted by Markus Scheuer.

Seiichi Manyama, Table of n, a(n) for n = 0..253
Arman Maesumi, Triangle Inscribed-Triangle Picking, arXiv:1804.11007 [math.GM], 2018.
R. Sprugnoli, Riordan Array Proofs of Identities in Gould's Book.

Cf. A000142, A001044, A003149, A100516, A100517.

Table[Sum[(k!*(n-k)!)^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 05 2016 *)

a(n) = Sum_{i=0..n} (i! * (n-i)!)^2.
a(n) ~ 2*(n!)^2. - Vaclav Kotesovec, Dec 05 2016
a(n) = A001044(n)*A100516(n)/A100517(n). - Alois P. Heinz, Feb 21 2023

Definition clarified by Georg Fischer, Feb 21 2023

A122184 Numerator of Sum_{k=0..2n} (-1)^k/C(2n,k)^3.

Original entry on oeis.org

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

Offset: 0

Alexander Adamchuk, May 10 2007

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

Eric Weisstein's World of Mathematics, Binomial Sums.

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.

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

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

A128152 Numerator of Sum_{k=0..n} 1/binomial(n,k)^4.

Original entry on oeis.org

1, 2, 33, 164, 20825, 10017, 25940593, 34743416, 3074035689, 672229195, 13443874324243, 431453199593, 53678600587865227, 33768054132971557, 813464644344955, 748569723383876272, 67454811525665973337193

Offset: 0

Alexander Adamchuk, May 10 2007

p^k divides a(p^k-1) for prime p and integer k > 0. p divides a(p-2) for prime p > 5.

Eric Weisstein's World of Mathematics, Binomial Sums.

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).

Table[ Numerator[ Sum[ 1 / Binomial[n,k]^4, {k,0,n} ] ], {n,0,50} ]

a(n) = numerator(Sum_{k=0..n} 1/binomial(n,k)^4).

cp's OEIS Frontend

A100518 Numerator of Sum_{k=0..n} 1/binomial(n,k)^3.

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

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A100519 Denominator of Sum_{k=0..n} 1/binomial(n,k)^3.

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A100520 Numerator of Sum_{k=0..2*n} (-1)^k/binomial(2*n, k)^2.

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A279055 Self-convolution of squares of factorial numbers (A001044).

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A122184 Numerator of Sum_{k=0..2n} (-1)^k/C(2n,k)^3.

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A128152 Numerator of Sum_{k=0..n} 1/binomial(n,k)^4.

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A100520 Numerator of Sum_{k=0..2n} (-1)^k/binomial(2n, k)^2.