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

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

Offset: 0

Author

Benoit Cloitre, Sep 08 2002

Keywords

• easy
• frac
• nonn

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

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

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