A136727 -id:A136727 - 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

A136728 E.g.f.: A(x) = (exp(x)/(4 - 3*exp(x)))^(1/4).

Original entry on oeis.org

1, 1, 4, 31, 349, 5146, 93799, 2036161, 51283894, 1470035101, 47250248569, 1683031711516, 65800765032589, 2801364476781781, 129003301751229364, 6389120632590635971, 338644807090096148809, 19126604338708282552186

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..373

Cf. A201354, variants: A014307, A136727, A136729.

CoefficientList[Series[(E^x/(4-3*E^x))^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 15 2013 *)

a(n)=n!*polcoeff((exp(x +x*O(x^n))/(4-3*exp(x +x*O(x^n))))^(1/4),n)

/* As solution to integral equation: */ a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+intformal(A^4*exp(-x+x*O(x^n))));n!*polcoeff(A,n)

E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^5 * exp(-x) ).
O.g.f.: 1/(1 - x/(1-3*x/(1 - 5*x/(1-6*x/(1 - 9*x/(1-9*x/(1 - 13*x/(1-12*x/(1 - 17*x/(1-15*x/(1 - ...))))))))))), a continued fraction.
G.f.: 1/G(0) where G(k) = 1 - x*(4*k+1)/( 1 - 3*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * Gamma(3/4)/(sqrt(2)*3^(1/4)*n^(3/4)*Pi*log(4/3)^(n+1/4)). - Vaclav Kotesovec, Jun 15 2013
a(n) = 1 + 3 * Sum_{k=1..n-1} (binomial(n,k) - 1) * a(k). - Ilya Gutkovskiy, Jul 09 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (4*j+1)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (3*k/n - 4) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 3*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

A136729 E.g.f.: A(x) = [ exp(x)/(5 - 4*exp(x)) ]^(1/5).

Original entry on oeis.org

1, 1, 5, 49, 701, 13177, 306821, 8520289, 274808525, 10095533833, 416131518293, 19017974164465, 954399901374749, 52173428322993433, 3085965087129209381, 196360349627069553793, 13374490368820471936109, 970904530181260115741737

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Variants: A014307, A136727, A136728.

CoefficientList[Series[(E^x/(5-4*E^x))^(1/5), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 22 2013 *)

a(n)=n!*polcoeff((exp(x +x*O(x^n))/(5-4*exp(x +x*O(x^n))))^(1/5),n)

/* As solution to integral equation: */ a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+intformal(A^6*exp(-x+x*O(x^n))));n!*polcoeff(A,n)

E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^6 * exp(-x) ).
O.g.f.: 1/(1 - x/(1-4*x/(1 - 6*x/(1-8*x/(1 - 11*x/(1-16*x/(1 - 16*x/(1-24*x/(1 - 21*x/(1-32*x/(1 - ...)))))))))), a continued fraction.
G.f.: 1/G(0) where G(k) = 1 - x*(5*k+1)/( 1 - 4*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * sqrt(5-sqrt(5))*Gamma(4/5) / (2*Log[5/4]^(n+1/5) * 2^(9/10)*n^(4/5)*Pi). - Vaclav Kotesovec, Sep 22 2013
a(n) = 1 + 4 * Sum_{k=1..n-1} (binomial(n,k) - 1) * a(k). - Ilya Gutkovskiy, Jul 09 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * (Product_{j=0..k-1} (5*j+1)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (4*k/n - 5) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 4*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

A185285 Triangle T(n,k), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 10, 6, 1, 0, 74, 52, 12, 1, 0, 730, 570, 160, 20, 1, 0, 9002, 7600, 2430, 380, 30, 1, 0, 133210, 119574, 42070, 7630, 770, 42, 1, 0, 2299754, 2170252, 822696, 166320, 19740, 1400, 56, 1, 0, 45375130, 44657106, 17985268, 3956568, 528780, 44604, 2352, 72, 1

Offset: 0

Philippe Deléham, Dec 22 2011

The Bell transform of A004123(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins :
1
0, 1
0, 2, 1
0, 10, 6, 1
0, 74, 52, 12, 1
0, 730, 570, 160, 20, 1
0, 9002, 7600, 2430, 380, 30, 1
0, 133210, 119574, 42070, 7630, 770, 42, 1

Row sums are A136727.

Cf. A000007, A004123, A000012, A002378, A195205.

(* The function BellMatrix is defined in A264428. *)
a4123[n_] := If[n == 1, 1, PolyLog[-n+1, 2/3]/3];
rows = 10;
M = BellMatrix[a4123[#+1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 25 2019 *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: A004123(n+1), 10) # Peter Luschny, Jan 18 2016

More terms from Jean-François Alcover, Jun 25 2019

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

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A136728 E.g.f.: A(x) = (exp(x)/(4 - 3*exp(x)))^(1/4).

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A136729 E.g.f.: A(x) = [ exp(x)/(5 - 4*exp(x)) ]^(1/5).

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A185285 Triangle T(n,k), read by rows, given by (0, 2, 3, 4, 6, 6, 9, 8, 12, 10, 15, ...) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938.

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A367371 Expansion of the e.g.f. (exp(x) / (3 - 2*exp(x)))^(2/3).

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