OFFSET
0,1
LINKS
Petros Hadjicostas, Table of n, a(n) for n = 0..101
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(2*Sum_{m=1..n+1} Sum_{p=0..m-1} (-1)^p * (m!/((p+1)*3^(m+2))) * Stirling2(n+1,m) * binomial(2*p,p) * binomial(m-1,p))). [It follows from Theorem 1 in Dyson et al. (2010-2011, 2013).] - Petros Hadjicostas, May 15 2020
MAPLE
LehmerSer := n -> 2*add(add((-1)^p*(m!/((p+1)*3^(m+2)))*Stirling2(n+1, m)
*binomial(2*p, p)*binomial(m-1, p), p=0..m-1), m=1..n+1):
a := n -> numer(LehmerSer(n)): seq(a(n), n=0..21);
# (after Petros Hadjicostas) Peter Luschny, May 15 2020
MATHEMATICA
f[n_] := Sum[j^n/Binomial[2*j, j], {j, 1, Infinity}];
a[n_] := Expand[ FunctionExpand[ f[n] ] ][[2, 1]] // Numerator;
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 24 2017 *)
PROG
(PARI) a(n)=numerator(2*sum(m=1, n+1, sum(p=0, m-1, (-1)^p*(m!/((p+1)*3^(m+2)))*stirling(n+1, m, 2)*binomial(2*p, p)*binomial(m-1, p)))) \\ Petros Hadjicostas, May 15 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 09 2011, following a suggestion from Herb Conn
EXTENSIONS
a(11)-a(21) from Nathaniel Johnston, Apr 07 2011
STATUS
approved