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 1-4 of 4 results.

A046825 Numerator of Sum_{k=0..n} 1/binomial(n,k).

Original entry on oeis.org

1, 2, 5, 8, 8, 13, 151, 256, 83, 146, 1433, 2588, 15341, 28211, 52235, 19456, 19345, 36362, 651745, 6168632, 1463914, 2786599, 122289917, 233836352, 140001721, 268709146, 774885169, 1491969394, 41711914513, 80530073893
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The term a(12)=15341 is divisible by 23^2. Is there another term a(n) divisible by the square of a prime p larger than n+1? - M. F. Hasler, Jul 17 2012

Examples

			1, 2, 5/2, 8/3, 8/3, 13/5, 151/60, 256/105, 83/35, 146/63, 1433/630, 2588/1155, 15341/6930, 28211/12870, 52235/24024, 19456/9009, 19345/9009, ... = A046825/A046826

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294, Problem 7.15.
  • R. L. Graham, D. E. Knuth, O. Patashnik; Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed. Exercise 5.100.
  • G. Letac, Problèmes de probabilitĂ©s, Presses Universitaires de France (1970), p. 14.
  • F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

Links

Crossrefs

Cf. A003149, A046826, A048211, A051389, A100512, A100513.

Programs

  • Magma
    [Numerator((&+[1/Binomial(n,j): j in [0..n]])): n in [0..40]]; // G. C. Greubel, May 24 2021
  • Mathematica
    Numerator/@Table[Sum[1/Binomial[n,k],{k,0,n}],{n,0,40}]  (* Harvey P. Dale, Apr 21 2011 *)
  • PARI
    P=1;vector(30,n,numerator(P)+0*P=P/2/n*(n+1)+1) \\ M. F. Hasler, Jul 17 2012
  • PARI
    A046825(n)=numerator(sum(k=0,n,1/binomial(n,k))) \\ M. F. Hasler, Jul 19 2012
  • Sage
    [numerator(sum(1/binomial(n,j) for j in (0..n))) for n in (0..40)] # G. C. Greubel, May 24 2021

Formula

Let P(n) = (1/n) * Sum_{k=0..n-1} 1/binomial(n-1, k) = A046878(n)/A046879(n) = A046825(n-1)/(n*A046826(n-1)): { 0, 1, 1, 5/6, 2/3, 8/15, ...}. Then P(n) = 2^(-n) * Sum_{k=1..n} 2^k / k = 2^(-n+1) * Sum_{k odd} binomial(n, k)/k; P(0) = 0, P(n) = P(n-1)/2 + 1/n. - Torsten Sillke (Torsten.Sillke(AT)uni-bielefeld.de)
G.f. for P(n): (2*log(1-z))/(-2+z). - Wouter Meeussen
P(n) = 2^(-n) * Sum_{k=1..n} (binomial(n,k) + 1)/k.
a(n) = numerator( A003149(n)/n! ). - G. C. Greubel, May 24 2021

Extensions

References entries (Comtet, Graham et al., Letac, Nedemeyer) and Links entries (Singmaster, Sury) from Torsten.Sillke(AT)uni-bielefeld.de

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

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, A046826, A100516, A100517, A100519.

Programs

  • Magma
    [Numerator( (&+[1/Binomial(n,k)^3: k in [0..n]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022
  • Mathematica
    Numerator[Table[Sum[1/Binomial[n,k]^3,{k,0,n}],{n,0,20}]] (* Harvey P. Dale, Sep 28 2012 *)
  • PARI
    a(n) = numerator(sum(k=0, n, 1/binomial(n,k)^3)); \\ Michel Marcus, Jun 24 2022
  • SageMath
    [numerator(sum(1/binomial(n,k)^3 for k in (0..n))) for n in (0..40)] # G. C. Greubel, Jun 24 2022

Formula

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

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

A354479 a(n) is the denominator of Sum_{k=1..n} 1 / Stirling1(n,k).

Original entry on oeis.org

1, 1, 6, 33, 4200, 4192200, 5115600, 545250747888, 10086416728304192640, 126556188275836361347200, 451535899566923284351392000, 250606479905655959999200124455664175360, 1493469115548888160803495265626573200563200, 47083781674990641531154175811928872812783834939059200
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 02 2022

Keywords

Examples

			1, 0, 7/6, 25/33, 3991/4200, 3923773/4192200, 4901627/5115600, 527165212865/545250747888, ...

Crossrefs

Cf. A008275, A046826, A112289, A112291, A354478 (numerators).

Programs

  • Mathematica
    Table[Sum[1/StirlingS1[n, k], {k, 1, n}], {n, 1, 14}] // Denominator
  • PARI
    a(n) = denominator(sum(k=1, n, 1/stirling(n, k, 1))); \\ Michel Marcus, Jun 02 2022
Showing 1-4 of 4 results.

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

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