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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.

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A278140 Numerator of partial sums of a Ramanujan series with value 2/(Gamma(3/4)^4), given in A277235.

Original entry on oeis.org

1, 27, 29835, 914095, 30845936835, 966228811317, 1005862016542383, 31766194302634935, 33673399154070922824435, 1067731823813513897297545, 1101976780048026596318593989, 35023352480137647877041347193, 1154564397329013014999165944225975
Offset: 0

Views

Author

Wolfdieter Lang, Nov 13 2016

Keywords

Comments

The denominators are given in A074800.
One of Ramanujan's series is 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 - 13*(1*3*5/(2*4*6))^5 +- ... = Sum_{k>=0} (-1)^k*(1+4*k)*(risefac(1/2,k)/k!)^5 where risefac(x,k) = Product_{j =0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 7, eq. (1.4) and pp. 105-106, 111. The value of this series is 2/(Gamma(3/4)^4) given in A277235.

Examples

			The rationals r(n) begin: 1, 27/32, 29835/32768, 914095/1048576, 30845936835/34359738368, 966228811317/1099511627776, 1005862016542383/1125899906842624, ...
The limit r(n), for n -> oo, is 2/(Gamma(3/4)^4) given in A277235.

References

  • G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106.

Crossrefs

Cf. A074799/A074800, A277235.

Formula

a(n) = numerator(r(n)), with the rationals r(n) = Sum_{k=0..n} (-1)^k*(1+4*k)*(risefac(1/2,k)/k!)^5 = 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 rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k)/A074800(k).

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
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