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A052841
Expansion of e.g.f.: 1/(exp(x)*(2-exp(x))).
53
1, 0, 2, 6, 38, 270, 2342, 23646, 272918, 3543630, 51123782, 811316286, 14045783798, 263429174190, 5320671485222, 115141595488926, 2657827340990678, 65185383514567950, 1692767331628422662, 46400793659664205566, 1338843898122192101558, 40562412499252036940910
OFFSET
0,3
COMMENTS
From Michael Somos, Mar 04 2004: (Start)
Stirling transform of A005359(n)=[0,2,0,24,0,720,...] is a(n)=[0,2,6,38,270,...].
Stirling transform of -(-1)^n*A052657(n-1)=[0,0,2,-6,48,-240,...] is a(n-1)=[0,0,2,6,38,270,...].
Stirling transform of -(-1)^n*A052558(n-1)=[1,-1,4,-12,72,-360,...] is a(n-1)=[1,0,2,6,38,270,...].
Stirling transform of 2*A052591(n)=[2,4,24,96,...] is a(n+1)=[2,6,38,270,...].
(End)
Also the central moments of a Geometric(1/2) random variable (for example the number of coin tosses until the first head). - Svante Janson, Dec 10 2012
Also the number of ordered set partitions of {1..n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n). - Gus Wiseman, Feb 13 2019
Also the number of ordered set partitions of {1..n} with an even number of blocks. - Geoffrey Critzer, Jul 04 2020
LINKS
Camilo Barreto, Pamela Harris, José L. Ramírez, Samuel Ramírez, and Julio C. Vasquez, Restricted Fubini Rankings and Restricted Unit Interval Parking Functions, arXiv:2511.01997 [math.CO], 2025. See pp. 10, 12.
Christian G. Bower, Transforms (2).
Svante Janson, Euler-Frobenius numbers and rounding, arXiv:1305.3512 [math.PR], 2013. See. p. 31.
Lukas Spiegelhofer, A lower bound for Cusick's conjecture on the digits of n+t, arXiv:1910.13170 [math.NT], 2019. See p. 6.
FORMULA
O.g.f.: Sum_{n>=0} (2*n)! * x^(2*n) / Product_{k=1..2*n} (1-k*x). - Paul D. Hanna, Jul 20 2011
a(n) = (A000670(n) + (-1)^n)/2 = Sum_{k>=0} (k-1)^n/2^(k+1). - Vladeta Jovovic, Feb 02 2003
Also, a(n) = Sum_{k=0..[n/2]} (2k)!*Stirling2(n, 2k). - Ralf Stephan, May 23 2004
a(n) = D^n*(1/(1-x^2)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000670 and A005649. - Peter Bala, Nov 25 2011
E.g.f.: 1/(2*G(0)), where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k*(k+1)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 22 2012
a(n) ~ n!/(4*(log(2))^(n+1)). - Vaclav Kotesovec, Aug 10 2013
a(n) = (h(n)+(-1)^n)/2 where h(n) = Sum_{k=0..n} E(n,k)*2^k and E(n,k) the Eulerian numbers A173018 (see also A156365). - Peter Luschny, Sep 19 2015
a(n) = (-1)^n + Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 11 2020
EXAMPLE
From Gus Wiseman, Feb 13 2019: (Start)
The a(4) = 38 ordered set partitions with no cyclical adjacencies:
{{1}{2}{3}{4}} {{1}{24}{3}} {{13}{24}}
{{1}{2}{4}{3}} {{1}{3}{24}} {{24}{13}}
{{1}{3}{2}{4}} {{13}{2}{4}}
{{1}{3}{4}{2}} {{13}{4}{2}}
{{1}{4}{2}{3}} {{2}{13}{4}}
{{1}{4}{3}{2}} {{2}{4}{13}}
{{2}{1}{3}{4}} {{24}{1}{3}}
{{2}{1}{4}{3}} {{24}{3}{1}}
{{2}{3}{1}{4}} {{3}{1}{24}}
{{2}{3}{4}{1}} {{3}{24}{1}}
{{2}{4}{1}{3}} {{4}{13}{2}}
{{2}{4}{3}{1}} {{4}{2}{13}}
{{3}{1}{2}{4}}
{{3}{1}{4}{2}}
{{3}{2}{1}{4}}
{{3}{2}{4}{1}}
{{3}{4}{1}{2}}
{{3}{4}{2}{1}}
{{4}{1}{2}{3}}
{{4}{1}{3}{2}}
{{4}{2}{1}{3}}
{{4}{2}{3}{1}}
{{4}{3}{1}{2}}
{{4}{3}{2}{1}}
(End)
MAPLE
spec := [S, {B=Prod(C, C), C=Set(Z, 1 <= card), S=Sequence(B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
P := proc(n, x) option remember; if n = 0 then 1 else
(n*x+2*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x); expand(%) fi end:
A052841 := n -> subs(x=2, P(n, x)):
seq(A052841(n), n=0..21); # Peter Luschny, Mar 07 2014
h := n -> add(combinat:-eulerian1(n, k)*2^k, k=0..n):
a := n -> (h(n)+(-1)^n)/2: seq(a(n), n=0..21); # Peter Luschny, Sep 19 2015
b := proc(n, m) option remember; if n = 0 then 1 else
(m - 1)*b(n - 1, m) + (m + 1)*b(n - 1, m + 1) fi end:
a := n -> b(n, 0): seq(a(n), n = 0..21); # Peter Luschny, Jun 23 2023
MATHEMATICA
a[n_] := If[n == 0, 1, (PolyLog[-n, 1/2]/2 + (-1)^n)/2]; (* or *)
a[n_] := HurwitzLerchPhi[1/2, -n, -1]/2; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 19 2016, after Vladeta Jovovic *)
With[{nn=30}, CoefficientList[Series[1/(Exp[x](2-Exp[x])), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Apr 08 2019 *)
PROG
(PARI) a(n)=if(n<0, 0, n!*polcoeff(subst(1/(1-y^2), y, exp(x+x*O(x^n))-1), n))
(PARI) {a(n)=polcoeff(sum(m=0, n, (2*m)!*x^(2*m)/prod(k=1, 2*m, 1-k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024
(SageMath)
def A052841_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( exp(-x)/(2-exp(x)) ).egf_to_ogf().list()
A052841_list(40) # G. C. Greubel, Jun 11 2024
CROSSREFS
Main diagonal of A122101.
Inverse binomial transform of A000670.
Sequence in context: A027322 A085447 A078673 * A275557 A197972 A068184
KEYWORD
easy,nonn
EXTENSIONS
Edited by N. J. A. Sloane, Sep 06 2013
STATUS
approved