A100517 -id:A100517 - cp's OEIS frontend

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

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, A100517, A100518.

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

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 *)

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

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

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

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

Original entry on oeis.org

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 ).

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 ).

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

cp's OEIS Frontend

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

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

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

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

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