A074800 -id:A074800 - cp's OEIS frontend

A074799 a(n) = numerator( (4n+1)(Product_{i=1..n} (2i-1)/Product_{i=1..n} 2i)^5 ).

1, 5, 2187, 40625, 892871875, 20841167403, 16443713753775, 421390226721321, 364130196991193221875, 9816949116755633084375, 8619392462988365485907909, 239904481399203205153660455

Offset: 0

Benoit Cloitre, Sep 08 2002

Bruce C. Berndt and Robert Rankin, "Ramanujan : letters and commentary", AMS-LMS, History of mathematics vol. 9, p. 57

Seiichi Manyama, Table of n, a(n) for n = 0..335

Cf. A074800 (denominators).

[Numerator((4*n+1)*((n+1)*Catalan(n)/4^n)^5): n in [0..30]]; // G. C. Greubel, Jul 09 2021

Table[Numerator[(4*n+1)*(Binomial[2*n, n]/4^n)^5], {n,0,30}] (* G. C. Greubel, Jul 09 2021 *)

a(n)=numerator((4*n+1)*(prod(i=1,n,2*i-1)/prod(i=1,n,2*i))^5)

[numerator((4*n+1)*(binomial(2*n, n)/4^n)^5) for n in (0..30)] # G. C. Greubel, Jul 09 2021

a(n) = numerator of (b(n)), where b(n) = (4*n+1)*(Product_{i=1..n} (2*i - 1)/Product_{i=1..n} 2*i)^5 and b(0) = 1.
1 + Sum_{k>=1} (-1)^k*b(k) = 2/gamma(3/4)^4 = 0.88694116857811540541...
a(n) = numerator( (4*n+1)*( binomial(2*n, n)/4^n )^5 ). - G. C. Greubel, Jul 09 2021

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

Wolfdieter Lang, Nov 13 2016

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.

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

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.

G. C. Greubel, Table of n, a(n) for n = 0..10000

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

SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // G. C. Greubel, Oct 26 2018

RealDigits[2/(Gamma[3/4])^4, 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)

2/gamma(3/4)^4 \\ Michel Marcus, Nov 13 2016

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)

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

Wolfdieter Lang, Nov 13 2016

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.

			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.

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

Cf. A074799/A074800, A277235.

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).

cp's OEIS Frontend

A074799 a(n) = numerator( (4n+1)(Product_{i=1..n} (2i-1)/Product_{i=1..n} 2i)^5 ).

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

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

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cp's OEIS Frontend

A074799 a(n) = numerator( (4*n+1)*(Product_{i=1..n} (2*i-1)/Product_{i=1..n} 2*i)^5 ).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Formula

A074799 a(n) = numerator( (4n+1)(Product_{i=1..n} (2i-1)/Product_{i=1..n} 2i)^5 ).