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

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

Offset: 0

Author

Benoit Cloitre, Sep 08 2002

Keywords

• easy
• frac
• nonn

References

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

Links

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

Crossrefs

Cf. A074800 (denominators).

Programs

Magma

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

Mathematica

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

PARI

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

Sage

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

Formula

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