A014307 -id:A014307 - cp's OEIS frontend

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

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

A195204 Triangle of coefficients of a sequence of binomial type polynomials.

Original entry on oeis.org

2, 2, 4, 6, 12, 8, 26, 60, 48, 16, 150, 380, 360, 160, 32, 1082, 2940, 3120, 1680, 480, 64, 9366, 26908, 31080, 19040, 6720, 1344, 128, 94586, 284508, 351344, 236880, 96320, 24192, 3584, 256

Offset: 1

Peter Bala, Sep 13 2011

Define a polynomial sequence P_n(x) by means of the recursion
P_(n+1)(x) = x*(P_n(x)+ P_n(x+1)), with P_0(x) = 1.
The first few polynomials are
P_1(x) = 2*x, P_2(x) = 2*x*(2*x + 1),
P_3(x) = 2*x*(4*x^2 + 6*x + 3), P_4(x) = 2*x*(8*x^3+24*x^2+30*x+13).
The present table shows the coefficients of these polynomials (excluding P_0(x)) in ascending powers of x. The P_n(x) are a polynomial sequence of binomial type. In particular, if we denote P_n(x) by x^[n] then we have the analog of the binomial expansion
(x+y)^[n] = Sum_{k = 0..n} binomial(n,k)*x^[n-k]*y^[k].
There are further analogies between the x^[n] and the monomials x^n.
1) Dobinski-type formula
exp(-x)*Sum_{k >= 0} (-k)^[n]*x^k/k! = (-1)^n*Bell(n,2*x),
where the Bell (or exponential) polynomials are defined as
Bell(n,x) := Sum_{k = 1..n} Stirling2(n,k)*x^k.
Equivalently, the connection constants associated with the polynomial sequences {x^[n]} and {x^n} are (up to signs) the same as the connection constants associated with the polynomial sequences {Bell(n,2*x)} and {Bell(n,x)}. For example, the list of coefficients of x^[4] is [26,60,48,16] and a calculation gives
Bell(4,2*x) = -26*Bell(1,x) + 60*Bell(2,x) - 48*Bell(3,x) + 16*Bell(4,x).
2) Analog of Bernoulli's summation formula
Bernoulli's formula for the sum of the p-th powers of the first n positive integers is
Sum_{k = 1..n} k^p = (1/(p+1))*Sum_{k = 0..p} (-1)^k * binomial(p+1,k)*B_k*n^(p+1-k), where B_k = [1,-1/2,1/6,0,-1/30,...] is the sequence of Bernoulli numbers.
This generalizes to
2*Sum_{k = 1..n} k^[p] = 1/(p+1)*Sum_{k = 0..p} (-1)^k * binomial(p+1,k)*B_k*n^[p+1-k].
The polynomials P_n(x) belong to a family of polynomial sequences P_n(x,t) of binomial type, dependent on a parameter t, and defined recursively by P_(n+1)(x,t)= x*(P_n(x,t)+ t*P_n(x+1,t)), with P_0(x,t) = 1. When t = 0 we have P_n(x,0) = x^n, the monomial polynomials. The present table is the case t = 1. The case t = -2 is (up to signs) A079641. See also A195205 (case t = 2).
Triangle T(n,k) (1 <= k <= n), read by rows, given by (0, 1, 2, 2, 4, 3, 6, 4, 8, 5, 10, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 22 2011
T(n,k) is the number of binary relations R on [n] with index = 1 containing exactly k strongly connected components (SCC's) and satisfying the condition that if (x,y) is in R then x and y are in the same SCC. - Geoffrey Critzer, Jan 17 2024

			Triangle begins
n\k|....1......2......3......4......5......6......7
===================================================
..1|....2
..2|....2......4
..3|....6.....12......8
..4|...26.....60.....48.....16
..5|..150....380....360....160.....32
..6|.1082...2940...3120...1680....480.....64
..7|.9366..26908..31080..19040...6720...1344....128
...
Relation with rising factorials for row 4:
x^[4] = 16*x^4+48*x^3+60*x^2+26*x = 2^4*x*(x+1)*(x+2)*(x+3)-6*2^3*x*(x+1)*(x+2)+7*2^2*x*(x+1)-2*x, where [1,7,6,1] is the fourth row of the triangle of Stirling numbers of the second kind A008277.
Generalized Dobinski formula for row 4:
exp(-x)*Sum_{k >= 1} (-k)^[4]*x^k/k! = exp(-x)*Sum_{k >= 1} (16*k^4-48*k^3+60*k^2-26*k)*x^k/k! = 16*x^4+48*x^3+28*x^2+2*x = Bell(4,2*x).
Example of generalized Bernoulli summation formula:
2*(1^[2]+2^[2]+...+n^[2]) = 1/3*(B_0*n^[3]-3*B_1*n^[2]+3*B_2*n^[1]) =
n*(n+1)*(4*n+5)/3, where B_0 = 1, B_1 = -1/2, B_2 = 1/6 are Bernoulli numbers.
From _Philippe Deléham_, Dec 22 2011: (Start)
Triangle (0, 1, 2, 2, 4, 3, 6, ...) DELTA (2, 0, 2, 0, 2, ...) begins:
  1;
  0,    2;
  0,    2,     4;
  0,    6,    12,     8;
  0,   26,    60,    48,    16;
  0,  150,   380,   360,   160,   32;
  0, 1082,  2940,  3120,  1680,  480,   64;
  0, 9366, 26908, 31080, 19040, 6720, 1344, 128;
  ... (End)

Cf. A000629 (row sums), A000670 (one half row sums), A014307 (row polys. at x = 1/2), A079641, A195205, A209849.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (-1)^(n+1)*polylog(-n, 2), 10); # Peter Luschny, Jan 29 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
M = BellMatrix[(-1)^(#+1) PolyLog[-#, 2]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 24 2018, after Peter Luschny *)

E.g.f.: F(x,z) := (exp(z)/(2-exp(z)))^x = Sum_{n>=0} P_n(x)*z^n/n!
= 1 + 2*x*z + (2*x+4*x^2)*z^2/2! + (6*x+12*x^2+8*x^3)*z^3/3! + ....
The generating function F(x,z) satisfies the partial differential equation d/dz(F(x,z)) = x*F(x,z) + x*F(x+1,z) and hence the row polynomials P_n(x) satisfy the recurrence relation
P_(n+1)(x)= x*(P_n(x) + P_n(x+1)), with P_0(x) = 1.
In what follows we change notation and write x^[n] for P_n(x).
Relation with the factorial polynomials:
For n >= 1,
x^[n] = Sum_{k = 1..n} (-1)^(n-k)*Stirling2(n,k)*2^k*x^(k),
and its inverse formula
2^n*x^(n) = Sum_{k = 1..n} |Stirling1(n,k)|*x^[k],
where x^(n) denotes the rising factorial x*(x+1)*...*(x+n-1).
Relation with the Bell polynomials:
The alternating n-th row entries (-1)^(n+k)*T(n,k) are the connection coefficients expressing the polynomial Bell(n,2*x) as a linear combination of Bell(k,x), 1 <= k <= n.
The delta operator:
The sequence of row polynomials is of binomial type. If D denotes the derivative operator d/dx then the delta operator D* for this sequence of binomial type polynomials is given by
D* = D/2 - log(cosh(D/2)) = log(2*exp(D)/(exp(D)+1))
= (D/2) - (D/2)^2/2! + 2*(D/2)^4/4! - 16*(D/2)^6/6! + 272*(D/2)^8/8! - ...,
where [1,2,16,272,...] is the sequence of tangent numbers A000182.
D* is the lowering operator for the row polynomials
(D*)x^[n] = n*x^[n-1].
Associated Bernoulli polynomials:
Generalized Bernoulli polynomial GB(n,x) associated with the polynomials x^[n] may be defined by
GB(n,x) := ((D*)/(exp(D)-1))x^[n].
They satisfy the difference equation
GB(n,x+1) - GB(n,x) = n*x^[n-1]
and have the expansion
GB(n,x) = -(1/2)*n*x^[n-1] + (1/2)*Sum_{k = 0..n} binomial(n,k) * B_k * x^[n-k], where B_k denotes the ordinary Bernoulli numbers.
The first few polynomials are
GB(0,x) = 1/2, GB(1,x) = x-3/4, GB(2,x) = 2*x^2-2*x+1/12,
GB(3,x) = 4*x^3-3*x^2-x, GB(4,x) = 8*x^4-4*x^2-4*x-1/60.
It can be shown that
1/(n+1)*(d/dx)(GB(n+1,x)) = Sum_{i = 0..n} 1/(i+1) * Sum_{k = 0..i} (-1)^k *binomial(i,k)*(x+k)^[n].
This generalizes a well-known formula for Bernoulli polynomials.
Relations with other sequences:
Row sums: A000629(n) = 2*A000670(n). Column 1: 2*A000670(n-1). Row polynomials evaluated at x = 1/2: {P_n(1/2)}n>=0 = [1,1,2,7,35,226,...] = A014307.
T(n,k) = A184962(n,k)*2^k. - Philippe Deléham, Feb 17 2013
Also the Bell transform of A076726. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016
Conjecture: o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 2*x*z/(1 - z/(1 - 2*(x + 1)*z/(1 - 2*z/(1 - 2*(x + 2)*z/(1 - 3*z/(1 - 2*(x + 3)*z/(1 - 4*z/(1 - ... ))))))))). - Peter Bala, Dec 12 2024

a(1) added by Philippe Deléham, Dec 22 2011

A337520 Number of set partitions of [4n] into 4-element subsets {i, i+k, i+2k, i+3k} with 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 4, 10, 22, 64, 147, 409, 1092, 3253, 8661, 28585, 83190, 274001, 912373, 3366384, 13253582, 61533277, 290493694

Offset: 0

Alois P. Heinz, Nov 18 2020

			a(4) = 10: {{1,2,3,4}, {5,6,7,8}, {9,10,11,12}, {13,14,15,16}},
  {{1,3,5,7}, {2,4,6,8}, {9,10,11,12}, {13,14,15,16}},
  {{1,2,3,4}, {5,7,9,11}, {6,8,10,12}, {13,14,15,16}},
  {{1,4,7,10}, {2,5,8,11}, {3,6,9,12}, {13,14,15,16}},
  {{1,2,3,4}, {5,6,7,8}, {9,11,13,15}, {10,12,14,16}},
  {{1,3,5,7}, {2,4,6,8}, {9,11,13,15}, {10,12,14,16}},
  {{2,4,6,8}, {1,5,9,13}, {3,7,11,15}, {10,12,14,16}},
  {{1,2,3,4}, {5,8,11,14}, {6,9,12,15}, {7,10,13,16}},
  {{1,3,5,7}, {2,6,10,14}, {9,11,13,15}, {4,8,12,16}},
  {{1,5,9,13}, {2,6,10,14}, {3,7,11,15}, {4,8,12,16}}.

Wikipedia, Partition of a set

Cf. A000566, A014307, A104430, A334250.

Main diagonal of A360333.

b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({seq(m-h*j, h=1..3)} minus s={}, b(s minus {seq(m-h*j,
      h=0..3)}, t), 0), j=1..min(t, iquo(m-1, 3))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..4*n}, n) end:
seq(a(n), n=0..12);

b[s_, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[       If[Union@Table[m-h*j, {h, 1, 3}] ~Complement~ s == {}, b[s ~Complement~ Union@Table[m-h*j, {h, 0, 3}], t], 0], {j, 1, Min[t, Quotient[m-1, 3]]}]][Max[s]]];
a[n_] := a[n] = b[Range[4n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 13 2023, after Alois P. Heinz *)

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

A211399 Triangle T(n,k), 0 <= k <= n, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 15, 18, 1, 0, 1, 37, 129, 58, 1, 0, 1, 83, 646, 877, 179, 1, 0, 1, 177, 2685, 8030, 5280, 543, 1, 0, 1, 367, 10002, 56285, 82610, 29658, 1636, 1, 0, 1, 749, 34777, 335162

Offset: 0

Philippe Deléham, Feb 08 2013

Contains A156920 as submatrix.

Row-reversal of A102365. - Philippe Deléham, Feb 12 2013

			Triangle begins :
1
0, 1
0, 1, 1
0, 1, 5, 1
0, 1, 15, 18, 1
0, 1, 37, 129, 58, 1
0, 1, 83, 646, 877, 179, 1

Left hand column sequences: A000007, A000012, A050488, A142965, A142966, A142968.
Right hand column sequences: A000340, A156922, A156923, A156924.
Row sums A014307(n).

T(n,k) = k*T(n-1,k) + (2n-2k+1)*T(n-1,k-1) , T(n,n) = 1, T(n,k) = 0 if k<0 or if k>n.
T(n,k) = A185411(n,k)/(2^(n-k)).
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000012(n), A014307(n), A001147(n) for x = 0, 1, 2 respectively .
G.f.: 1/(1-xy/(1-x/(1-3xy/(1-2x/(1-5xy/(1-3x/(1-7xy/(1- ...(continued fraction).

A334250 Number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1<=k<=n.

Original entry on oeis.org

1, 1, 2, 4, 12, 35, 129, 567, 2920, 16110, 103467, 717608, 5748214, 47937957, 441139750, 4319093093, 45963368076, 510202534002, 6150655137844, 76789781005325, 1028853084775725, 14294680087131380

Offset: 0

Alois P. Heinz, Apr 20 2020

Differs from A331621 first at n=7.

			a(2) = 2: 123|456, 135|246.
a(3) = 4: 123|456|789, 123|468|579, 135|246|789, 147|258|369.

Wikipedia, Partition of a set

Cf. A014307 (the same for 2-element subsets), A025035, A059108, A104429 (where k is not restricted), A285527, A331621, A337520.

Main diagonal of A360334.

b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({m-j, m-2*j} minus s={}, b(s minus {m, m-j, m-2*j},
            t), 0), j=1..min(t, iquo(m-1, 2))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..3*n}, n) end:
seq(a(n), n=0..12);

b[s_List, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[If[{m - j, m - 2j} ~Complement~ s == {}, b[s ~Complement~ {m, m - j, m - 2j}, t], 0], {j, 1, Min[t, Quotient[m - 1, 2]]}]][Max[s]]];
a[n_] := a[n] = b[Range[3n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 12}] (* Jean-François Alcover, May 10 2020, after Maple *)

a(n) <= A104429(n) <= A025035(n).

a(17)-a(21) from Martin Fuller, Jul 19 2025

A348468 Expansion of e.g.f. sqrt(exp(x)*(2-exp(x))).

Original entry on oeis.org

1, 0, -1, -3, -10, -45, -271, -2058, -18775, -199335, -2410516, -32683563, -490870315, -8087188200, -144994236661, -2810079139143, -58536519252130, -1304198088413265, -30946462816602331, -779104979758256298, -20742005411397108595, -582214473250983046155, -17184302765073000634276

Offset: 0

Michel Marcus, Oct 19 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..425
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.

Cf. A000629, A000918, A014304, A014307, A260701, A260902.

m = 22; Range[0, m]! * CoefficientList[Series[Sqrt[Exp[x]*(2 - Exp[x])], {x, 0, m}], x] (* Amiram Eldar, Oct 19 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(sqrt(exp(x)*(2-exp(x)))))

a(n) ~ -sqrt(2) * n^(n-1) / (log(2)^(n - 1/2) * exp(n)). - Vaclav Kotesovec, Oct 21 2021

A009383 Expansion of log(1+tanh(log(1+x))).

Original entry on oeis.org

0, 1, -2, 5, -15, 54, -240, 1350, -9450, 78120, -725760, 7371000, -81081000, 965487600, -12454041600, 173675502000, -2605132530000, 41763850128000, -711374856192000, 12817252047600000, -243527788904400000

Offset: 0

R. H. Hardin

Also expansion of e.g.f. log(1/(1 + Sum_{k>=1} (k+1)/2 * (-x)^k)). - Seiichi Manyama, Jun 01 2019

			log(1/(1 + Sum_{k>=1} (k+1)/2 * (-x)^k)) = x - 2*x^2/2! + 5*x^3/3! - 15*x^4/4! + 54*x^5/5! - 240*x^6/6! + 1350*x^7/7! - 9450*x^8/8! + ... . - _Seiichi Manyama_, Jun 01 2019

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A014307.

With[{nn=30},CoefficientList[Series[Log[1+Tanh[Log[1+x]]],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jan 27 2012 *)
CoefficientList[Series[Log[2 - 2/(2 + x*(2 + x))], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2015 *)

{a(n) = if (n<1, 0, -((-1)^n*(n+1)!+sum(k=1, n-1, binomial(n-1, k)*(-1)^k*(k+1)!*a(n-k)))/2)} \\ Seiichi Manyama, Jun 01 2019

The e.g.f. equals log(2(x+1)^2/(x^2+2x+2)), which has compositional inverse sqrt(exp(x)/(2-exp(x))) - 1. See A014307. - Peter Bala, Mar 23 2013
a(n) ~ 2 * (n-1)! * (-1)^(n+1). - Vaclav Kotesovec, Jan 23 2015
a(n) = (-1/2) * ((-1)^n * (n+1)! + Sum_{k=1..n-1} binomial(n-1,k) * (-1)^k * (k+1)! * a(n-k)). - Seiichi Manyama, Jun 01 2019

Extended with signs by Olivier Gérard, Mar 15 1997

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

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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A195204 Triangle of coefficients of a sequence of binomial type polynomials.

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A337520 Number of set partitions of [4n] into 4-element subsets {i, i+k, i+2k, i+3k} with 1<=k<=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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A211399 Triangle T(n,k), 0 <= k <= n, given by (0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...) DELTA (1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...) where DELTA is the operator defined in A084938.

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A334250 Number of set partitions of [3n] into 3-element subsets {i, i+k, i+2k} with 1<=k<=n.

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A348468 Expansion of e.g.f. sqrt(exp(x)*(2-exp(x))).

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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.