A259852 - OEIS

%I #25 May 14 2020 21:47:30

%S 1,8,18,128,200,192,784,8192,10368,25600,30976,147456,173056,401408,

%T 10240,8388608,9469952,7077888,23658496,20971520,38535168,253755392,

%U 277348352,268435456,2621440000,5670699008,6115295232,3758096384,28219277312,60397977600

%N 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).

%H F. J. Dyson, N. E. Frankel, and M. L. Glasser, <a href="http://arxiv.org/abs/1009.4274">Lehmer's Interesting Series</a>, arXiv:1009.4274 [math-ph], 2010-2011.

%H F. J. Dyson, N. E. Frankel, and M. L. Glasser, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.120.02.116">Lehmer's interesting series</a>, Amer. Math. Monthly, 120 (2013), 116-130.

%H D. H. Lehmer, <a href="https://www.jstor.org/stable/2322496">Interesting series involving the central binomial coefficient</a>, Amer. Math. Monthly, 92(7) (1985), 449-457.

%F a(n) = numerator(n^2*2^n/C(2*n,n)).

%F 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]

%e 1/1, 8/3, 18/5, 128/35, 200/63, 192/77, 784/429, ... = A259852/A259853.

%t Table[2^n*n^2/Binomial[2*n, n] // Numerator, {n, 1, 40}]

%o (PARI) vector(40, n, numerator(n^2*2^n/binomial(2*n,n))) \\ _Michel Marcus_, Jul 07 2015

%Y Cf. A014307, A180875, A259853 (denominators).

%K nonn,frac,easy

%O 1,2

%A _Jean-François Alcover_, Jul 07 2015