A278142 -id:A278142 - cp's OEIS frontend

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

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

Offset: 0

Wolfdieter Lang, Nov 14 2016

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)).

			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... .

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

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A278142, A278146.

Numerator[Table[ Sum[  (1 + 8*k)*(Binomial[-1/4, k])^4 , {k, 0, n}] , {n, 0, 25}]] (* G. C. Greubel, Jan 09 2017 *)

for(n=0,10, print1( numerator( sum(k=0,n, (1+8*k)*(binomial(-1/4,k))^4)), ", ")) \\ G. C. Greubel, Jan 09 2017

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.

A278146 Decimal expansion of 2^(3/2) / (sqrt(Pi)*Gamma(3/4)^2).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 15 2016

This is the value of a series of Ramanujan, namely 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 (after division by s).

For the partial sums of this series see A278141/A278142.

			1.06267989991684365118249019510...

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

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A278141/A278142, A242439.

First@ RealDigits@ N[2^(3/2)/(Sqrt[Pi] Gamma[3/4]^2), 104] (* Michael De Vlieger, Nov 15 2016 *)
RealDigits[2^(3/2)/(Sqrt[Pi]*(Gamma[3/4])^2), 10, 50][[1]] (* G. C. Greubel, Jan 10 2017 *)

2^(3/2)/(sqrt(Pi)*(gamma(3/4))^2) \\ G. C. Greubel, Jan 10 2017

2^(3/2) / (sqrt(Pi)*Gamma(3/4)^2).

Equals 2*A242439. - Hugo Pfoertner, Apr 26 2025

A381268 a(n) = denominator( [(xyzu)^n] 1/sqrt(1 - (x + y + z + u(y + z))) ).

Original entry on oeis.org

1, 4, 256, 1024, 1048576, 4194304, 268435456, 1073741824, 17592186044416, 70368744177664, 4503599627370496, 18014398509481984, 18446744073709551616, 73786976294838206464, 4722366482869645213696, 18889465931478580854784, 4951760157141521099596496896, 19807040628566084398385987584

Offset: 0

Stefano Spezia, Feb 18 2025

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 24.

Cf. A381267 (numerator).

Cf. A009971, A134375, A278142.

a[n_]:=Denominator[SeriesCoefficient[1/Sqrt[1-(x+y+z+u(y+z))],{x,0,n},{y,0,n},{z,0,n},{u,0,n}]]; Array[a,13,0]

a(n) = denominator( [x^n] hypergeom( [1/2, 1/6, 1/2, 5/6], [1, 1, 1], 108*x) ).
a(n) = denominator( 2^(2*n-1) * 27^n * Gamma(n+1/6) * Gamma(n+1/2)^2 * Gamma(n+5/6)/(Pi^2 * (n!)^4) ).
a(2*n) = A278142(n).

cp's OEIS Frontend

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

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A278146 Decimal expansion of 2^(3/2) / (sqrt(Pi)*Gamma(3/4)^2).

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A381268 a(n) = denominator( [(xyzu)^n] 1/sqrt(1 - (x + y + z + u(y + z))) ).

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cp's OEIS Frontend

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

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Formula

A278146 Decimal expansion of 2^(3/2) / (sqrt(Pi)*Gamma(3/4)^2).

Views

Author

Keywords

Comments

Examples

References

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Crossrefs

Programs

Mathematica

PARI

Formula

A381268 a(n) = denominator( [(x*y*z*u)^n] 1/sqrt(1 - (x + y + z + u*(y + z))) ).

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Formula

A381268 a(n) = denominator( [(xyzu)^n] 1/sqrt(1 - (x + y + z + u(y + z))) ).