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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).
2
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
1,2
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.
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]
EXAMPLE
1/1, 8/3, 18/5, 128/35, 200/63, 192/77, 784/429, ... = A259852/A259853.
MATHEMATICA
Table[2^n*n^2/Binomial[2*n, n] // Numerator, {n, 1, 40}]
PROG
(PARI) vector(40, n, numerator(n^2*2^n/binomial(2*n, n))) \\ Michel Marcus, Jul 07 2015
CROSSREFS
Cf. A014307, A180875, A259853 (denominators).
Sequence in context: A113563 A173734 A219524 * A215174 A171371 A092692
KEYWORD
nonn,frac,easy
AUTHOR
STATUS
approved