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 1-3 of 3 results.

A241756 A finite sum of products of binomial coefficients: Sum_(m=0..n) binomial(-1/4, m)^2*binomial(-1/4, n-m)^2 (C. C. Grosjean's problem, denominators).

Original entry on oeis.org

1, 8, 512, 4096, 2097152, 16777216, 1073741824, 8589934592, 35184372088832, 281474976710656, 18014398509481984, 144115188075855872, 73786976294838206464, 590295810358705651712, 37778931862957161709568, 302231454903657293676544
Offset: 0

Views

Author

Jean-François Alcover, Apr 28 2014

Keywords

Comments

This sequence seems to appear also as denominators of A277232, A277234, and A278143. - Wolfdieter Lang, Nov 16 2016

References

  • E. S. Andersen and M. E. Larsen. A finite sum of products of binomial coefficients, Problem 92-18, by C. C. Grosjean, Solution. SIAM Rev. 35 (1993), 645-646.

Links

Crossrefs

Cf. A241755 (numerators), A277232, A277234, A278143.

Programs

  • Mathematica
    a[n_] := Binomial[2*n, n]^2*Binomial[n-1/2, 2*n]*(-1/4)^n; Table[a[n]//Denominator, {n, 0, 20}]

Formula

GAMMA(3/4)^2 * 4F3(1/4, 1/4, -n, -n; 1, 3/4-n, 3/4-n; 1)/(GAMMA(3/4-n)^2*GAMMA(n+1)^2).
binomial(2n, n)^2*binomial(n-1/2, 2n)*(-1/4)^n.
Conjecture (from sequencedb.net): a(n) = 8^A005187(n). - R. J. Mathar, Jun 30 2021

A278141 Numerators of partial sums of a Ramanujan series converging to 2^(3/2)/(sqrt(Pi)*Gamma(3/4)^2) given in A278146.

Original entry on oeis.org

1, 265, 1096065, 281858265, 18519577975665, 4748934018906441, 19474365987782658225, 4989739877102195271225, 5235591401647346852339166225, 1341015791319444602368386319225, 5495144390631448939048252704196225, 1407253983507773608409169421000239225, 92253220393640211712365553562313715740225
Offset: 0

Views

Author

Wolfdieter Lang, Nov 14 2016

Keywords

Comments

The denominators are given in A278142.
One of Ramanujan's series is 1 + 9*(1/4)^4 + 17*(1*5/(4*8))^4 + 25*(1*5*9/(4*8*12))^4 + ... = Sum_{k>=0} (1+8*k)*(risefac(1/4,k)/k!)^4 where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 7, eq. (1.3) and p. 105, eq. (7.4.3) for s=1/4. The value of this series is 2^(3/2)/(sqrt(Pi)*Gamma(3/4)^2) given in A278146.
The general formula, Hardy, p. 105, eq. (7.4.3) (divided by s) is Sum_{k>=0} (1 + 2*k/s)*(risefac(s,k)/k!)^4 = sin^2(s*Pi)*Gamma(s)^2/(2*s*Pi^2*cos(s*Pi)* Gamma(2*s)).

Examples

			The rationals begin: 1, 265/256, 1096065/1048576, 281858265/268435456, 18519577975665/17592186044416, 4748934018906441/4503599627370496, 19474365987782658225/18446744073709551616, ...
The value of the series is (see A278143)
  2^(3/2)/(sqrt(Pi)*Gamma(3/4)^2) = 1.06267989991... .

References

  • G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105.

Links

Crossrefs

Cf. A278142, A278146.

Programs

  • Mathematica
    Numerator[Table[ Sum[  (1 + 8*k)*(Binomial[-1/4, k])^4 , {k, 0, n}] , {n, 0, 25}]] (* G. C. Greubel, Jan 09 2017 *)
  • PARI
    for(n=0,10, print1( numerator( sum(k=0,n, (1+8*k)*(binomial(-1/4,k))^4)), ", ")) \\ G. C. Greubel, Jan 09 2017

Formula

a(n) = numerator(r(n)), with the rationals r(n) = Sum_{k=0..n} (1+8*k)*(risefac(1/4,k)/k!)^4. The rising factorial has been defined in a comment above.
a(n) = Sum_{k=0..n} (1+8*k)*(binomial(-1/4,k))^4.

A278144 Decimal expansion of (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2.

Original entry on oeis.org

9, 0, 9, 1, 7, 2, 7, 9, 4, 5, 4, 6, 9, 2, 9, 7, 0, 0, 7, 3, 9, 7, 7, 8, 8, 5, 4, 2, 8, 2, 6, 5, 1, 2, 2, 5, 7, 2, 0, 5, 2, 7, 2, 9, 9, 5, 9, 2, 2, 0, 5, 2, 2, 8, 3, 8, 6, 4, 1, 4, 0, 2, 1, 8, 3, 7, 2, 2, 3, 6, 4, 8, 1, 1, 1, 2, 7, 1, 8, 9, 9, 3, 2, 3, 2, 5, 6, 7, 4, 0, 5, 7, 0, 5, 1, 3, 7, 9, 5, 3, 3, 7, 3
Offset: 0

Views

Author

Wolfdieter Lang, Nov 14 2016

Keywords

Comments

This is the value of hypergeometric([1/4,1/4],[1],-1)^2. See A278143/A241756 for the partial sums of the hypergeometric series hypergeometric([1/2/,1/2,1/2],[1,1],-1) which has this value due to Clausen's formula. See the Hardy reference, p. 106, eq. (7.4.4) where this value is written as (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2.

Examples

			The value of the series 1 - (1/2)^3 + (1*3/(2*4))^3 - (1*3*5/(2*4*6))^3 + ... is 0.909172794546929700739778854282651225720527299592205228386414021837...
This is also the value of the series Sum_{n>=0} c(n) with c(n) = Sum_{k=0..n} f(k)*f(n-k), where f(0)=1 and f(k) = (-1)^k*(1*5*9 *** (4*k-3)/(4*8*12 *** (4*k)))^2, k >= 1 (self-convolution of the hypergeometric([1/4,1/4],[1],-1) series).

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
  • G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 106, eq. (7.4.4)

Links

Crossrefs

Cf. A278143.

Programs

  • Magma
    pi:=Pi(RealField(110)); (Sqrt(pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2; // Felix Fröhlich, Nov 15 2016
  • Mathematica
    RealDigits[(Pi/Sqrt[2])*(1/(Gamma[5/8]*Gamma[7/8]))^2, 10, 50][[1]] (* G. C. Greubel, Jan 12 2017 *)
  • PARI
    (sqrt(Pi)/(2^(1/4)*gamma(5/8)*gamma(7/8)))^2 \\ Felix Fröhlich, Nov 15 2016

Formula

Equals hypergeometric([1/2/,1/2,1/2],[1,1],-1) = hypergeometric([1/4,1/4],[1],-1)^2 = Sum_{k>=0} (-1)^k*(risefac(k,1/2)/k!)^3, where risefac(x,m) = Product_{j =0..m-1} (x+j), and risefac(x,0) = 1.
Equals (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2 = (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2.
Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)^3/64^k. - Amiram Eldar, Jul 04 2023
Equals Gamma(1/8)^4 * (2 - sqrt(2)) / (16 * Pi^2 * Gamma(1/4)^2). - Vaclav Kotesovec, Jul 04 2023
Showing 1-3 of 3 results.

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