A346208 -id:A346208 - cp's OEIS frontend

A052841 Expansion of e.g.f.: 1/(exp(x)*(2-exp(x))).

1, 0, 2, 6, 38, 270, 2342, 23646, 272918, 3543630, 51123782, 811316286, 14045783798, 263429174190, 5320671485222, 115141595488926, 2657827340990678, 65185383514567950, 1692767331628422662, 46400793659664205566, 1338843898122192101558, 40562412499252036940910

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

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

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..200
C. G. Bower, Transforms (2)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 808
Svante Janson, Euler-Frobenius numbers and rounding, preprint arXiv:1305.3512 [math.PR], 2013.
Lukas Spiegelhofer, A lower bound for Cusick's conjecture on the digits of n+t, arXiv:1910.13170 [math.NT], 2019.

Main diagonal of A122101.
Inverse binomial transform of A000670.
Cf. A000126, A000296, A000670, A001610, A005359, A032032, A052558.
Cf. A052591, A052657, A052841, A124323, A156365, A169985, A173018.
Cf. A187784, A324011, A344037, A346208.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024

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

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 *)

a(n)=if(n<0,0,n!*polcoeff(subst(1/(1-y^2),y,exp(x+x*O(x^n))-1),n))

{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 */

def A052841_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(x)) ).egf_to_ogf().list()
A052841_list(40) # G. C. Greubel, Jun 11 2024

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

Edited by N. J. A. Sloane, Sep 06 2013

A344037 Expansion of e.g.f.: exp(-2*x) / (2 - exp(x)).

Original entry on oeis.org

1, -1, 3, -1, 27, 119, 1203, 11759, 136587, 1771559, 25562403, 405657119, 7022893947, 131714582999, 2660335750803, 57570797728079, 1328913670528107, 32592691757218439, 846383665814342403, 23200396829831840639, 669421949061096575067, 20281206249626017421879

Offset: 0

Ilya Gutkovskiy, Aug 01 2021

sign

G. C. Greubel, Table of n, a(n) for n = 0..420

Cf. A000670, A007047, A008619, A052841, A330603, A346208.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-2*x)/(2-Exp(x)) ))); // G. C. Greubel, Jun 11 2024

nmax = 21; CoefficientList[Series[Exp[-2 x]/(2 - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
Table[HurwitzLerchPhi[1/2, -n, -2]/2, {n, 0, 21}]
a[n_] := a[n] = (-2)^n + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 21}]

def A344037_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-2*x)/(2-exp(x)) ).egf_to_ogf().list()
A344037_list(40) # G. C. Greubel, Jun 11 2024

a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A000670(k).
a(n) = Sum_{k=0..n} (-1)^k * Stirling2(n,k) * k! * A008619(k).
a(n) = Sum_{k>=0} (k - 2)^n / 2^(k+1).
a(n) = (-2)^n + Sum_{k=0..n-1} binomial(n,k) * a(k).
a(n) ~ n! / (8 * log(2)^(n+1)). - Vaclav Kotesovec, Aug 15 2021

A367979 Expansion of e.g.f. exp(-x) / (2 - exp(3*x)).

Original entry on oeis.org

1, 2, 22, 278, 4822, 104342, 2709622, 82092278, 2842418902, 110720079062, 4792059271222, 228144844817078, 11849163703935382, 666694458859845782, 40397145162583154422, 2622634244645856386678, 181615748103175019442262, 13362823095925278064444502, 1041037845089466806646007222

Offset: 0

Ilya Gutkovskiy, Dec 07 2023

nonn

G. C. Greubel, Table of n, a(n) for n = 0..360

Cf. A000670, A052841, A151919, A328182, A346208, A367977, A367980, A367981, A367982, A367983.

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!(Laplace( Exp(-x)/(2-Exp(3*x)) ))); // G. C. Greubel, Jun 11 2024

nmax = 18; CoefficientList[Series[Exp[-x]/(2 - Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-1)^n + Sum[Binomial[n, k] 3^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]

def A367979_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-x)/(2-exp(3*x)) ).egf_to_ogf().list()
A367979_list(40) # G. C. Greubel, Jun 11 2024

a(n) = Sum_{k>=0} (3*k-1)^n / 2^(k+1).
a(n) = (-1)^n + Sum_{k=1..n} binomial(n,k) * 3^k * a(n-k).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * 3^k * A000670(k).

A367983 Expansion of e.g.f. exp(-3x) / (2 - exp(4x)).

Original entry on oeis.org

1, 1, 33, 481, 11457, 329281, 11405793, 460726561, 21270068097, 1104703800961, 63750028379553, 4046761389279841, 280235644230863937, 21023317859012763841, 1698493239420829750113, 147024466409751282556321, 13575133989036437786590977, 1331764937006253524751217921

Offset: 0

Ilya Gutkovskiy, Dec 07 2023

nonn

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A000670, A328183, A346208, A355220, A367977, A367979, A367980, A367981, A367982.

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(-3*x)/(2-Exp(4*x)) ))); // G. C. Greubel, Jun 11 2024

nmax = 17; CoefficientList[Series[Exp[-3 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (-3)^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

def A367983_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(-3*x)/(2-exp(4*x)) ).egf_to_ogf().list()
A367983_list(40) # G. C. Greubel, Jun 11 2024

a(n) = Sum_{k>=0} (4*k-3)^n / 2^(k+1).
a(n) = (-3)^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * 4^k * A000670(k).

cp's OEIS Frontend

A052841 Expansion of e.g.f.: 1/(exp(x)*(2-exp(x))).

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A344037 Expansion of e.g.f.: exp(-2*x) / (2 - exp(x)).

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A367979 Expansion of e.g.f. exp(-x) / (2 - exp(3*x)).

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A367983 Expansion of e.g.f. exp(-3*x) / (2 - exp(4*x)).

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A367983 Expansion of e.g.f. exp(-3x) / (2 - exp(4x)).