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 1-2 of 2 results.

A074800 a(n) = denominator( (4*n+1)*(Product_{i=1..n} (2*i-1)/Product_{i=1..n} (2*i))^5 ).

Original entry on oeis.org

1, 32, 32768, 1048576, 34359738368, 1099511627776, 1125899906842624, 36028797018963968, 37778931862957161709568, 1208925819614629174706176, 1237940039285380274899124224, 39614081257132168796771975168
Offset: 0

Views

Author

Benoit Cloitre, Sep 08 2002

Keywords

Comments

For the partial sums of the series given in the formula section see A278140(n)/a(n). The value of the series is given in A277235. - Wolfdieter Lang, Nov 15 2016

References

  • Bruce C. Berndt and Robert Rankin,"Ramanujan: letters and commentary", AMS-LMS, History of Mathematics, vol. 9, p. 57
  • G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.

Links

Crossrefs

Cf. A074799, A277235, A278140.

Programs

  • Magma
    [Denominator((4*n+1)*((n+1)*Catalan(n)/4^n)^5): n in [0..30]]; // G. C. Greubel, Jul 09 2021
  • Mathematica
    Table[Denominator[(4n+1) (Product[(2i-1), {i, n}]/Product[2i, {i, n}])^5], {n, 0, 10}] (* Michael De Vlieger, Nov 15 2016 *)
  • PARI
    a(n)=denominator ((4*n+1)*(prod(i=1,n,2*i-1)/prod(i=1,n,2*i))^5)
  • Sage
    [denominator((4*n+1)*(binomial(2*n, n)/4^n)^5) for n in (0..30)] # G. C. Greubel, Jul 09 2021

Formula

a(n) = denominator(b(n)) with b(0) = 1 and b(n) = (4*n+1)*(Product_{i=1..n} (2*i-1) / Product_{i=1..n}(2*i))^5 = (4*n+1)*(A001147(n)/A000165(n))^5.
1 + Sum_{k>=1} (-1)^k*b(k) = 2/Gamma(3/4)^4=0.88694116857811540541...(see
A277235).
a(n) = denominator( (4*n+1)*( binomial(2*n, n)/4^n )^5 ). - G. C. Greubel, Jul 09 2021

Extensions

Edited. - Wolfdieter Lang, Nov 15 2016

A277235 Decimal expansion of 2/(Gamma(3/4))^4.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Nov 13 2016

Keywords

Comments

This is the value of one of Ramanujan's series: 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 -13*(1*3*5/(2*4*6))^5 + - ... . See the Hardy reference p.7. eq. (1.4) and pp. 105-106. For the partial sums see A278140.
The proof of Hardy and Whipple mentioned in the Hardy reference reduces this series to (2/Pi)*Morley's series (for m=1/2). For this series see A277232 and A091670.

Examples

			2/Gamma(3/4)^4 = 0.88694116857811540541152...

References

  • G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.

Links

Crossrefs

Cf. A014549, A060294, A074799, A074800, A088538, A091670, A263809, A277232, A278140.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // G. C. Greubel, Oct 26 2018
  • Mathematica
    RealDigits[2/(Gamma[3/4])^4, 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)
  • PARI
    2/gamma(3/4)^4 \\ Michel Marcus, Nov 13 2016

Formula

Equals Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^5 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^5. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
Equals A060294 * A091670.
For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k) / A074800(k).
From Amiram Eldar, Jul 13 2023: (Start)
Equals (Gamma(1/4)/Pi)^4/2.
Equals A088538 * A014549^2.
Equals A263809/Pi. (End)
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