A181374 -id:A181374 - cp's OEIS frontend

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.

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

cp's OEIS Frontend

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.

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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.

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

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cp's OEIS Frontend

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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Maple

Mathematica

PARI

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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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Maple

Mathematica

PARI

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Extensions

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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Mathematica

Formula

Extensions

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.