A136728 -id:A136728 - 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

A201354 Expansion of e.g.f. exp(x) / (4 - 3*exp(x)).

Original entry on oeis.org

1, 4, 28, 292, 4060, 70564, 1471708, 35810212, 995827420, 31153998244, 1082931514588, 41407678132132, 1727226633730780, 78051253062575524, 3798351192214837468, 198049421007186054052, 11014905131587945490140, 650903915009792820650404, 40726453234725158535472348

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 4*x + 28*x^2/2! + 292*x^3/3! + 4060*x^4/4! + 70564*x^5/5! + ...
O.g.f.: A(x) = 1 + 4*x + 28*x^2 + 292*x^3 + 4060*x^4 + 70564*x^5 + ...
where A(x) = 1 + 4*x/(1+x) + 2!*4^2*x^2/((1+x)*(1+2*x)) + 3!*4^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*4^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...

G. C. Greubel, Table of n, a(n) for n = 0..370

Cf. A000629, A032033, A050352, A123227, A136728.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( 1/(4*Exp(-x) -3) ))); // G. C. Greubel, Jun 08 2020

seq(coeff(series(1/(4*exp(-x) -3), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Jun 08 2020

Table[Sum[(-1)^(n-k)*4^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)

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

{a(n)=polcoeff(sum(m=0, n, 4^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

{Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, (-1)^(n-k)*4^k*Stirling2(n, k)*k!)}

[sum( (-1)^(n-j)*4^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 08 2020

O.g.f.: A(x) = Sum_{n>=0} n! * 4^n*x^n / Product_{k=0..n} (1+k*x).
O.g.f.: A(x) = 1/(1 - 4*x/(1-3*x/(1 - 8*x/(1-6*x/(1 - 12*x/(1-9*x/(1 - 16*x/(1-12*x/(1 - 20*x/(1-15*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-1)^(n-k) * 4^k * Stirling2(n,k) * k!.
a(n) = 4*A050352(n) for n>0.
a(n) = Sum_{k=0..n} A123125(n,k)*4^k*3^(n-k). - Philippe Deléham, Nov 30 2011
a(n) = log(4/3) * Integral_{x = 0..oo} (ceiling(x))^n * (4/3)^(-x) dx. - Peter Bala, Feb 06 2015
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 6*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) ~ n! / (3*(log(4/3))^(n+1)). - Vaclav Kotesovec, Jun 13 2013
a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(0) = 1; a(n) = -4*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) + 3*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
a(n) = (4/3)*A032033(n) - (1/3)*0^n. - Seiichi Manyama, Dec 21 2023

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)

A136727 E.g.f.: A(x) = (exp(x)/(3 - 2*exp(x)))^(1/3).

Original entry on oeis.org

1, 1, 3, 17, 139, 1481, 19443, 303297, 5480219, 112549881, 2589274883, 65957355377, 1842897053099, 56038776055081, 1842278768795923, 65109900167188257, 2461735422517374779, 99148196540813749081

Offset: 0

Paul D. Hanna, Jan 24 2008

nonn

G.f. of variant A014307 is B(x) = sqrt(exp(x)/(2-exp(x))), which satisfies: B(x) = 1 + integral(B(x)^3*exp(-x)).

			E.g.f.: A(x) = 1 + x + 3/2*x^2 + 17/6*x^3 + 139/24*x^4 + 1481/120*x^5 +...

Harvey P. Dale, Table of n, a(n) for n = 0..350

Cf. A201339, variants: A014307, A136728, A136729.

With[{nn=20},CoefficientList[Series[(Exp[x]/(3-2Exp[x]))^(1/3),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 26 2013 *)

{a(n) = n!*polcoeff((exp(x +x*O(x^n))/(3-2*exp(x +x*O(x^n))))^(1/3),n)}
for(n=0,25,print1(a(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)}
for(n=0,25,print1(a(n),", "))

E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^4 * exp(-x) ).
O.g.f.: 1/(1 - x/(1-2*x/(1 - 4*x/(1-4*x/(1 - 7*x/(1-6*x/(1 - 10*x/(1-8*x/(1 - 13*x/(1-10*x/(1 - ...))))))))))), a continued fraction.
G.f.: 1/G(0) where G(k) = 1 - x*(3*k+1)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013
a(n) ~ n! * sqrt(3)*2^(2/3)*Gamma(2/3)/(4*Pi*n^(2/3)*(log(3/2))^(n+1/3)). - Vaclav Kotesovec, Jun 25 2013
a(n) = 1 + 2 * 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} (3*j+1)) * Stirling2(n,k).
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^k * (2*k/n - 3) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = a(n-1) + 2*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)

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

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A201354 Expansion of e.g.f. exp(x) / (4 - 3*exp(x)).

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

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A136727 E.g.f.: A(x) = (exp(x)/(3 - 2*exp(x)))^(1/3).

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A367372 Expansion of the e.g.f. (exp(x) / (4 - 3*exp(x)))^(1/2).

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A367373 Expansion of the e.g.f. (exp(x) / (4 - 3*exp(x)))^(3/4).

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