cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Nov 25 2004

Keywords

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

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

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

Views

Author

N. J. A. Sloane, Nov 25 2004

Keywords

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

Crossrefs

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

Programs

  • Magma
    [Denominator( (&+[1/Binomial(n,k)^2: k in [0..n]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022
  • Mathematica
    Table[Sum[1/Binomial[n,k]^2,{k,0,n}],{n,0,30}]//Denominator (* Harvey P. Dale, Apr 01 2019 *)
  • PARI
    a(n) = denominator(sum(k=0, n, 1/binomial(n,k)^2)); \\ Michel Marcus, 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

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

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

Views

Author

N. J. A. Sloane, Nov 25 2004

Keywords

Examples

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

Links

Crossrefs

Cf. A046825, A100516, A100517, A100518.

Programs

  • Magma
    [Denominator( (&+[1/Binomial(n,k)^3: k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, Jun 24 2022
  • Mathematica
    Table[Denominator[Sum[1/Binomial[n,k]^3, {k,0,n}]], {n,0,30}] (* G. C. Greubel, Jun 24 2022 *)
  • PARI
    a(n) = denominator(sum(k=0, n, 1/binomial(n,k)^3)); \\ Michel Marcus, Jun 25 2022
  • SageMath
    [denominator(sum(1/binomial(n,k)^3 for k in (0..n))) for n in (0..30)] # G. C. Greubel, Jun 24 2022

Formula

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

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

Views

Author

Alexander Adamchuk, May 10 2007

Keywords

Comments

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

Links

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.

Programs

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

Formula

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

Views

Author

Alexander Adamchuk, May 10 2007

Keywords

Comments

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

Links

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).

Programs

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

Formula

a(n) = numerator(Sum_{k=0..n} 1/binomial(n,k)^4).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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