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A014307
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Expansion of the e.g.f. sqrt(exp(x) / (2 - exp(x))).
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27
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1, 1, 2, 7, 35, 226, 1787, 16717, 180560, 2211181, 30273047, 458186752, 7596317885, 136907048461, 2665084902482, 55726440112987, 1245661569161135, 29642264728189066, 748158516941653967, 19962900431638852297, 561472467839585937560, 16602088291822017588121
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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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.)
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FORMULA
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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) dt satisfies B'(x) = tan(B(x)) + sec(B(x)). (End)
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
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MAPLE
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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):
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MATHEMATICA
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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 *)
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PROG
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(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
(GAP) 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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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