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A014307 Expansion of sqrt( exp(x) / ( 2 - exp(x) )). 21
1, 1, 2, 7, 35, 226, 1787, 16717, 180560, 2211181, 30273047, 458186752, 7596317885, 136907048461, 2665084902482, 55726440112987, 1245661569161135, 29642264728189066, 748158516941653967, 19962900431638852297 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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(2j+1,j+1/2)), with j integral. So we have a(n) = Sum_{j >= 0} (j+1/2)^(n-1)*rho(j+1/2). - Roland Groux, 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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

P. 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 Second Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011. (See Table 2)

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.

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.

FORMULA

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( A(x)^3 * exp(-x) ). - Paul D. Hanna, Jan 24 2008

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)).

E.g.f. B(x) = integral_{t = 0..x} A(t) 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))) = series reversion (integral_{t = 0..x} (sec(t)-tan(t))) = 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

MATHEMATICA

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 *)

PROG

(PARI) {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

(PARI) /* 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

(Maxima)

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 */

(MAGMA) m:=20; R<x>:=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

(Sage) 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

CROSSREFS

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

Sequence in context: A317421 A292182 A185054 * A000154 A003713 A058129

Adjacent sequences:  A014304 A014305 A014306 * A014308 A014309 A014310

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 24 04:08 EDT 2019. Contains 325290 sequences. (Running on oeis4.)