A180875 -id:A180875 - cp's OEIS frontend

A014307 Expansion of the e.g.f. sqrt(exp(x) / (2 - exp(x))).

1, 1, 2, 7, 35, 226, 1787, 16717, 180560, 2211181, 30273047, 458186752, 7596317885, 136907048461, 2665084902482, 55726440112987, 1245661569161135, 29642264728189066, 748158516941653967, 19962900431638852297, 561472467839585937560, 16602088291822017588121

Offset: 0

N. J. A. Sloane

nonn

The Hankel transform of this sequence is A121835. - Philippe Deléham, Aug 31 2006
a(n) is the moment of order (n-1) for the discrete measure associated to the weight rho(j + 1/2) = 2^(j + 1/2)/(Pi*binomial(2*j + 1, j + 1/2)), with j integral. So we have a(n) = Sum_{j >= 0} (j + 1/2)^(n-1)*rho(j + 1/2). - Groux Roland, Jan 05 2009
Let f(n) = Sum_{j >= 1} j^n*2^j/binomial(2*j, j) = r_n*Pi/2 + s_n; sequence gives r_{n-1}. For example, f(0) through f(5) are [1 + (1/2)*Pi, 3 + Pi, 11 + (7/2)*Pi, 55 + (35/2)*Pi, 355 + 113*Pi, 2807 + (1787/2)*Pi]. For s_n, see A180875. - N. J. A. Sloane, following a suggestion from Herb Conn, Feb 08 2011
Ren gives seven combinatorial interpretations for this sequence. - Peter Bala, Feb 01 2013
Number of left-right arrangements of [n] [Crane, 2015]. - N. J. A. Sloane, Nov 21 2014
In Dyson et al. (2010-2011, 2013), we have S_n(2) = Sum_{j>=1} j^n*2^j/binomial(2*j, j) = A014307(n+1)*Pi/2 + A180875(n) for n >= 1 (and S_0(2) is not defined). This series was originally defined by Lehmer (1985). - Petros Hadjicostas, May 14 2020

Seiichi Manyama, Table of n, a(n) for n = 0..424 (terms 0..100 from Vincenzo Librandi)
Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.
D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions, arXiv:math/0501052 [math.CA], 2005.
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011. (See Table IV on p. 14.)
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130. (See Table 2.)
M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.
Feng Qi and Mark Daniel Ward, Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function, arXiv:2110.08576 [math.CO], 2021.
Q. Ren, Ordered partitions and drawings of rooted plane trees arXiv:1301.6327 [math.CO], 2013-2014.
Andrew T. Wilson, Torus link homology and the nabla operator, arXiv preprint arXiv:1606.00764 [cond-mat.str-el], 2016.

Cf. A000110, A180875.
Variants: A136727, A136728, A136729. A190392.
Row sums of triangle A156920 (row sums (n) = a(n+1)). - Johannes W. Meijer, Feb 20 2009

Concatenation([1], List([1..20], n-> Sum([1..n], k-> Sum([k..n], m-> Stirling2(n,m)*Factorial(m)*Binomial(m-1,k-1)*Binomial(2*k-2,k-1)*(-2)^(1-k)/k )))); # G. C. Greubel, Oct 20 2019

m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/Sqrt(2*Exp(-x)-1) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Jun 30 2019

seq(coeff(series(1/sqrt(2*exp(-x)-1), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Oct 20 2019
a := n -> add((-1)^(n-k)*Stirling2(n,k)*doublefactorial(2*k-1), k=0..n):
seq(a(n), n = 0..21); # Peter Luschny, Oct 19 2021

a[n_] := Sum[ Sum[ StirlingS2[n, k]*k!*Binomial[k-1, m-1], {k, m, n}]/m*Binomial[2*m-2, m-1]*(-1)^(m-1)/2^(m-1), {m, 1, n}]; a[0]=1; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Sep 10 2012, after Vladimir Kruchinin *)
CoefficientList[Series[Sqrt[E^x/(2-E^x)], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 07 2014 *)
A014307 = ConstantArray[0,20]; A014307[[1]]=1; Do[A014307[[n+1]] = 1 + Sum[(-1+Binomial[n+1,j])*A014307[[j]],{j,1,n}],{n,1,19}]; Flatten[{1,A014307}] (* Vaclav Kotesovec after Jon Perry, Jan 07 2014 *)

a(n):=sum(sum(stirling2(n,k)*k!*binomial(k-1,m-1),k,m,n)/(m)* binomial(2*m-2,m-1)*(-1)^(m-1)/2^(m-1),m,1,n); /* Vladimir Kruchinin, May 10 2011 */

{a(n)=n!*polcoeff((exp(x +x*O(x^n))/(2-exp(x +x*O(x^n))))^(1/2),n)} \\ Paul D. Hanna, Jan 24 2008

/* As solution to integral equation: */ {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+intformal(A^3*exp(-x+x*O(x^n))));n!*polcoeff(A,n)} \\ Paul D. Hanna, Jan 24 2008

m = 20; T = taylor(1/sqrt(2*exp(-x)-1), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, Jun 30 2019

a(n+1) = 1 + Sum_{j=1..n} (-1 + binomial(n+1,j))*a(j). - Jon Perry, Apr 25 2005, corrected by Vaclav Kotesovec, Jan 07 2014
The Hankel transform of this sequence is A121835. - Philippe Deléham, Aug 31 2006
E.g.f. A(x) satisfies A(x) = 1 + Integral_{t=0..x} (A(t)^3 * exp(-t)) dt. - Paul D. Hanna, Jan 24 2008 [Edited by Petros Hadjicostas, May 14 2020]
From Vladimir Kruchinin, May 10 2011: (Start)
a(n) = Sum_{m=1..n} (Sum_{k=m..n} Stirling2(n,k)*k!*binomial(k-1,m-1))*(1/m)*binomial(2*m-2,m-1)*(-1)^(m-1)/2^(m-1), n > 0.
E.g.f. B(x) = Integral_{t = 0..x} A(t) dt satisfies B'(x) = tan(B(x)) + sec(B(x)). (End)
From Peter Bala, Aug 25 2011: (Start)
It follows from Vladimir Kruchinin's formula above that
Sum_{n>=1} a(n-1)*x^n/n! = series reversion (Integral_{t = 0..x} 1/(sec(t)+tan(t)) dt) = series reversion (Integral_{t = 0..x} (sec(t)-tan(t)) dt) = series reversion (x - x^2/2! + x^3/3! - 2*x^4/4! + 5*x^5/5! - 16*x^6/6! + ...) = x + x^2/2! + 2*x^3/3! + 7*x^4/4! + 35*x^5/5! + 226*x^6/6! + ....
Let f(x) = sec(x) + tan(x). Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = (d/dx)(f(x)*D^n[f](x)) for n >= 0 (see A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x)). Then by [Dominici, Theorem 4.1] we have a(n) = D^n[f](0). Compare with A190392.
(End)
G.f.: 1/G(0) where G(k) = 1 - x*(2*k+1)/( 1 - x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ sqrt(2) * n^n / (exp(n) * (log(2))^(n+1/2)). - Vaclav Kotesovec, Jan 07 2014
G.f.: R(0)/(1-x), where R(k) = 1 - x^2*(k+1)*(2*k+1)/(x^2*(k+1)*(2*k+1) - (3*x*k+x-1)*(3*x*k+4*x-1)/R(k+1)); (continued fraction). - Sergei N. Gladkovskii, Jan 30 2014
a(0) = 1 and a(n) = a(n-1) + Sum_{k=1..n-1} binomial(n-1, k-1)*a(k) for n > 0. - Seiichi Manyama, Oct 20 2019
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*(2*k-1)!! (see Qi/Ward). - Peter Luschny, Oct 19 2021
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (k/n - 2) * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 15 2023
Conjecture from Mikhail Kurkov, Jun 24 2025: (Start)
a(n) = R(n,0,2) where
  R(0,0,m) = 1,
  R(n,0,m) = Sum_{j=0..n-1} R(n-1,j,m),
  R(n,k,m) = m*R(n,0,m) - Sum_{j=0..k-1} R(n-1,j,m) for 0 < k <= n.
More generally, R(n,0,m) gives expansion of the e.g.f. (exp(x) / (m - (m-1)*exp(x)))^(1/m) for any m>0. (End)

Name edited by Petros Hadjicostas, May 14 2020

A129063 Third column (m=2) of triangle A129062 and second column of triangle A079641.

Original entry on oeis.org

1, 6, 36, 250, 2040, 19334, 209580, 2562354, 34915680, 524986462, 8636859924, 154357103498, 2978418173640, 61718786864310, 1367098836863868, 32236969655283682, 806313056758966320, 21322699350055313678, 594440128269066768612, 17424632249851351374906

Offset: 0

Wolfdieter Lang, May 04 2007

Seiichi Manyama, Table of n, a(n) for n = 0..422

A000629 gives second, resp. first column of A129062, resp. A079641.

Cf. A180875.

N=20; x='x+O('x^N); f=exp(x)/(2-exp(x)); Vec(serlaplace(deriv(f*intformal(f)))) \\ Seiichi Manyama, Oct 22 2019

a(n) = A129062(n+2,2), n>=0.
a(n) = A079641(n+1,1), n>=0.
E.g.f.: (d^2/dx^2)((-log(2-exp(x)))^2)/2.
E.g.f.: d/dx (f(x) * Integral f(x) dx), where f(x) = exp(x)/(2-exp(x)), cf. A000629. - Seiichi Manyama, Oct 22 2019
a(n) ~ n! * n * log(n) / (log(2))^(n+2) * (1 + (gamma - log(2) - log(log(2))) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 22 2019

A181374 Let f(n) = Sum_{j>=1} j^n3^j/binomial(2j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives s_n.

Original entry on oeis.org

3, 18, 156, 1890, 29496, 563094, 12709956, 331109658, 9777612432, 322738005150, 11775245575836, 470571509329506, 20441566147934568, 959052902557542246, 48330130399621041396, 2603558645653906065834, 149306059777139762896704, 9081311859252750219451182, 583927964165576868953730636

Offset: 0

N. J. A. Sloane, Feb 09 2011, following a suggestion from Herb Conn

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
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.

Cf. A185672 (r_n), A180875 and A014307 (2^j rather than 3^j).

f[n_] := Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}];
a[n_] := FindIntegerNullVector[{Pi/Sqrt[3], 1, N[-f[n], 20]}][[2]];
Table[s = a[n]; Print[s]; s, {n, 0, 8}] (* Jean-François Alcover, Sep 05 2018 *)
Table[Expand[FunctionExpand[FullSimplify[Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}]]]][[1]], {n, 0, 20}] (* Vaclav Kotesovec, May 14 2020 *)
S[k_, z_] := Sum[n!*(z/(4 - z))^n* StirlingS2[k + 1, n]*(1/n + Sum[(-1)^p*Pochhammer[1/2, p]/(p + 1)!* Binomial[n - 1, p]*(4/z)^(p + 1)*(Sqrt[z/(4 - z)]*ArcSin[Sqrt[z]/2] - 1/2*Sum[Gamma[l]/Pochhammer[1/2, l]*(z/4)^l, {l, 1, p}]), {p, 0, n - 1}]), {n, 1, k + 2}]; Table[Expand[Simplify[S[j, 3]]][[1]], {j, 0, 20}] (* Vaclav Kotesovec, May 15 2020 *)

a(n) ~ sqrt(2) * Pi * n^(n+1) / (3 * exp(n) * (log(4/3))^(n + 3/2)). - Vaclav Kotesovec, May 15 2020

More terms from Vaclav Kotesovec, May 14 2020

A181334 Let f(n) = Sum_{j>=1} j^n/binomial(2j,j) = r_nPi*sqrt(3)/3^{t_n} + s_n/3; sequence gives r_n.

Original entry on oeis.org

2, 2, 10, 74, 238, 938, 13130, 23594, 1298462, 26637166, 201403930, 5005052234, 135226271914, 1315508114654, 13747435592810, 153590068548062, 202980764290906, 69141791857625242, 2766595825017102650, 38897014541363246798, 1724835471991750464238, 80219728936311383557694

Offset: 0

N. J. A. Sloane, Feb 09 2011, following a suggestion from Herb Conn

nonn

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.

Cf. A098830 (s_n), A185585 (t_n), A181374, A180875, A014307.

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

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 *)

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

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

a(11)-a(21) from Nathaniel Johnston, Apr 07 2011

A185585 Let f(n) = Sum_{j>=1} j^n/binomial(2j,j) = r_nPi*sqrt(3)/3^{t_n} + s_n/3; sequence gives t_n.

Original entry on oeis.org

3, 3, 4, 5, 5, 5, 6, 5, 7, 8, 8, 9, 10, 10, 10, 10, 8, 11, 12, 12, 13, 14, 14, 13, 15, 13, 16, 17, 17, 18, 19, 19, 19, 20, 19, 21, 22, 22, 23, 24, 24, 24, 24, 23, 24, 25, 25, 26, 27, 27, 26, 28, 26, 29, 30, 30, 31, 32

Offset: 0

N. J. A. Sloane, Feb 09 2011, following a suggestion from Herb Conn

nonn

Petros Hadjicostas, Table of n, a(n) for n = 0..300
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.

Cf. A098830 (s_n), A181334 (r_n), A181374, A180875, A014307.

# The function LehmerSer is defined in A181334.
a := n -> ilog[3](denom(LehmerSer(n))):
seq(a(n), n=0..57); # Peter Luschny, May 15 2020

f[n_] := Sum[j^n/Binomial[2*j, j], {j, 1, Infinity}];
a[n_] := 1 + Log[3, Denominator[Expand[FunctionExpand[f[n]]][[2, 1]]]];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 60}] (* Jean-François Alcover, Nov 24 2017 *)

a(n) = logint(denominator(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)))), 3) \\ Petros Hadjicostas, May 14 2020

a(n) = ilog[3](denominator(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))), where ilog[3](3^k) = k. [It follows from Theorem 1 in Dyson et al. (2013).] - Petros Hadjicostas, May 14 2020

a(11)-a(57) from Nathaniel Johnston, Apr 07 2011

A185672 Let f(n) = Sum_{j>=1} j^n3^j/binomial(2j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives r_n.

Original entry on oeis.org

4, 20, 172, 2084, 32524, 620900, 14014732, 365100644, 10781360524, 355869575780, 12984066273292, 518879340911204, 22540052170064524, 1057507154836226660, 53291594817628483852, 2870834224548449841764, 164633490033421041392524, 10013579272685278891133540, 643872718978606529940390412

Offset: 0

N. J. A. Sloane, Feb 09 2011, following a suggestion from Herb Conn

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
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.

Cf. A181374 (s_n), A180875 and A014307 (2^j rather than 3^j).

f[n_] := Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}];
a[n_] := FindIntegerNullVector[{Pi/Sqrt[3], 1, N[-f[n], 20]}][[1]];
Table[r = a[n]; Print[r]; r, {n, 0, 8}] (* Jean-François Alcover, Sep 05 2018 *)
Table[Expand[FunctionExpand[FullSimplify[Sum[j^n*3^j/Binomial[2*j, j], {j, 1, Infinity}]]]][[2]] * Sqrt[3]/Pi, {n, 0, 20}] (* Vaclav Kotesovec, May 14 2020 *)
S[k_, z_] := Sum[n!*(z/(4 - z))^n* StirlingS2[k + 1, n]*(1/n + Sum[(-1)^p*Pochhammer[1/2, p]/(p + 1)!* Binomial[n - 1, p]*(4/z)^(p + 1)*(Sqrt[z/(4 - z)]*ArcSin[Sqrt[z]/2] - 1/2*Sum[Gamma[l]/Pochhammer[1/2, l]*(z/4)^l, {l, 1, p}]), {p, 0, n - 1}]), {n, 1, k + 2}]; Table[Expand[Simplify[S[j, 3]]][[2]]*Sqrt[3]/Pi, {j, 0, 20}] (* Vaclav Kotesovec, May 15 2020 *)

a(n) ~ 2^(3/2) * n^(n+1) / (sqrt(3) * exp(n) * (log(4/3))^(n + 3/2)). - Vaclav Kotesovec, May 15 2020

More terms from Vaclav Kotesovec, May 14 2020

A259852 Numerators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^kx^n/binomial(2n, n).

Original entry on oeis.org

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

Jean-François Alcover, Jul 07 2015

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

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.

Cf. A014307, A180875, A259853 (denominators).

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

vector(40, n, numerator(n^2*2^n/binomial(2*n,n))) \\ Michel Marcus, Jul 07 2015

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]

A259853 Denominators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^kx^n/binomial(2n, n).

Original entry on oeis.org

1, 3, 5, 35, 63, 77, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 215441, 300540195, 583401555, 756261275, 4418157975, 6892326441, 22427411435, 263012370465, 514589420475, 895766768975, 15801325804719, 61989816618513, 121683714103007

Offset: 1

Jean-François Alcover, Jul 07 2015

The first 14 terms are identical to A052468.

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

F. J. Dyson, N. E. Frankel, 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.

Cf. A014307, A052468, A180875, A259852 (numerators).

Table[2^n*n^2/Binomial[2*n, n] // Denominator, {n, 1, 40}]

vector(40, n, denominator(n^2*2^n/binomial(2*n,n))) \\ Michel Marcus, Jul 07 2015

a(n) = denominator(n^2*2^n/C(2*n,n)).

cp's OEIS Frontend

A014307 Expansion of the e.g.f. sqrt(exp(x) / (2 - exp(x))).

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A129063 Third column (m=2) of triangle A129062 and second column of triangle A079641.

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A181374 Let f(n) = Sum_{j>=1} j^n*3^j/binomial(2*j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives s_n.

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A181334 Let f(n) = Sum_{j>=1} j^n/binomial(2*j,j) = r_n*Pi*sqrt(3)/3^{t_n} + s_n/3; sequence gives r_n.

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A185585 Let f(n) = Sum_{j>=1} j^n/binomial(2*j,j) = r_n*Pi*sqrt(3)/3^{t_n} + s_n/3; sequence gives t_n.

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A185672 Let f(n) = Sum_{j>=1} j^n*3^j/binomial(2*j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives r_n.

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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).

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A181374 Let f(n) = Sum_{j>=1} j^n3^j/binomial(2j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives s_n.

A181334 Let f(n) = Sum_{j>=1} j^n/binomial(2j,j) = r_nPi*sqrt(3)/3^{t_n} + s_n/3; sequence gives r_n.

A185585 Let f(n) = Sum_{j>=1} j^n/binomial(2j,j) = r_nPi*sqrt(3)/3^{t_n} + s_n/3; sequence gives t_n.

A185672 Let f(n) = Sum_{j>=1} j^n3^j/binomial(2j,j) = r_n*Pi/sqrt(3) + s_n; sequence gives r_n.

A259852 Numerators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^kx^n/binomial(2n, n).

A259853 Denominators of the terms of Lehmer's series S_2(2), where S_k(x) = Sum_{n>=1} n^kx^n/binomial(2n, n).

A382629 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = (n-k)T(n-1,k-1) + 2(k+1)*T(n-1,k) + A102365(n,k) with T(n,k) = 0 if k < 0 or k > n.