%I #116 Jun 12 2023 02:36:57

%S 1,3,11,55,355,2807,26259,283623,3473315,47552791,719718067,

%T 11932268231,215053088835,4186305575415,87534887434835,

%U 1956680617267879,46561960552921315,1175204650272267479,31357650670190565363,881958890078887314567,26078499305918584929155,808742391638178302137783

%N Sum_{j>=1} j^n*2^j/binomial(2*j,j) = r_n*Pi/2 + s_n with integer r_n and s_n; sequence gives s_n.

%C In the references, the infinite series is S_n(2) = A014307(n+1)*Pi/2 + A180875(n) for n >= 1 (and S_0(2) is not defined). - _Petros Hadjicostas_, May 14 2020

%H Seiichi Manyama, <a href="/A180875/b180875.txt">Table of n, a(n) for n = 0..423</a>

%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; see Table IV on p. 14.

%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; see Table 2.

%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.

%H Feng Qi and Mark Daniel Ward, <a href="https://arxiv.org/abs/2110.08576">Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function</a>, arXiv:2110.08576 [math.CO], 2021.

%F a(0)=1; if n>=1, then a(n) = a(n-1) + 1 + Sum_{m=1..n} binomial(n,m)*a(n-m). - _Detlef Meya_, Jan 22 2018

%F E.g.f.: 2*(arcsin(exp(x/2)/sqrt(2)) - Pi/4) * sqrt(exp(x)/(2-exp(x))^3) + exp(x)/(2-exp(x)). - _Seiichi Manyama_, Oct 21 2019

%F a(n) ~ Pi * n^(n+1) / (sqrt(2) * exp(n) * (log(2))^(n + 3/2)). - _Vaclav Kotesovec_, Oct 22 2019

%F E.g.f.: d/dx (f(x) * Integral f(x) dx), where f(x) = sqrt(exp(x)/(2-exp(x))), cf. A014307. - _Seiichi Manyama_, Oct 22 2019

%p f := n -> sum(j^n*(j!)^2*2^j/(2*j)!, j = 1..infinity):

%p seq(f(n), n = 0..5); # gives

%p # [1+(1/2)*Pi, 3+Pi, 11+(7/2)*Pi, 55+(35/2)*Pi, 355+113*Pi, 2807+(1787/2)*Pi].

%t Table[Expand[FunctionExpand[FullSimplify[Sum[j^n*2^j/Binomial[2*j, j], {j, 1, Infinity}]]]][[1]], {n, 0, 20}] (* _Vaclav Kotesovec_, May 14 2020 *)

%o (PARI) N=20; x='x+O('x^N); f=sqrt(exp(x)/(2-exp(x))); Vec(serlaplace(deriv(f*intformal(f)))) \\ _Seiichi Manyama_, Oct 22 2019

%o (Python) # An alternative version of the sequence starts (for n >= 0):

%o # 0, 1, 3, 11, ..., or in terms of the approximation: [(1/2)*Pi, 1+(1/2)*Pi,

%o # 3+Pi, 11+(7/2)*Pi, ...]. Similar to the formula of _Detlef Meya_ above, the

%o # sequence then can be computed (without a special initial case) as:

%o from functools import cache

%o from math import comb as binomial

%o @cache

%o def a(n): return n + sum((binomial(n, j) - 1) * a(n - j) for j in range(1, n))

%o print([a(n) for n in range(23)]) # _Peter Luschny_, Jun 09 2023

%Y The values of r_n give A014307.

%Y Cf. A000629, A129063.

%K nonn,easy

%O 0,2

%A _Jonathan Vos Post_, Sep 23 2010

%E Attribution corrected by _M. Lawrence Glasser_, Sep 25 2010

%E Provided a better definition following a suggestion from _Herb Conn_. - _N. J. A. Sloane_, Feb 08 2011

%E Missing a(15) inserted by _Seiichi Manyama_, Oct 20 2019