A259852 Numerators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^k*x^n/binomial(2*n, n).
1, 8, 18, 128, 200, 192, 784, 8192, 10368, 25600, 30976, 147456, 173056, 401408, 10240, 8388608, 9469952, 7077888, 23658496, 20971520, 38535168, 253755392, 277348352, 268435456, 2621440000, 5670699008, 6115295232, 3758096384, 28219277312, 60397977600
Offset: 1
Examples
1/1, 8/3, 18/5, 128/35, 200/63, 192/77, 784/429, ... = A259852/A259853.
Links
- F. J. Dyson, N. E. Frankel, and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011.
- F. J. Dyson, N. E. Frankel, and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130.
- D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.
Programs
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Mathematica
Table[2^n*n^2/Binomial[2*n, n] // Numerator, {n, 1, 40}] -
PARI
vector(40, n, numerator(n^2*2^n/binomial(2*n,n))) \\ Michel Marcus, Jul 07 2015
Formula
a(n) = numerator(n^2*2^n/C(2*n,n)).
S_2(2) = Sum_{n>=1} 2^n*n^2/binomial(2*n, n) = 3F2([2,2,2]; [1,3/2]; 1/2) = 11 + 7*Pi/2. [Corrected by Petros Hadjicostas, May 14 2020]