A091346 -id:A091346 - cp's OEIS frontend

A083410 a(n) = A083385(n)/n.

1, 4, 22, 154, 1306, 12994, 148282, 1908274, 27333706, 431220034, 7428550042, 138737478994, 2792050329706, 60231133487074, 1386484468239802, 33921605427779314, 878976357571495306, 24046780495646314114, 692622345890928153562, 20950628198687114521234, 663992311200423614606506

Offset: 1

N. J. A. Sloane, Jun 08 2003

nonn

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of A052849(n+1)=[4,12,48,240,...] is 4*a(n)=[4,16,88,616,...].
Stirling transform of A001710(n+1)=[1,3,12,160,...] is a(n)=[1,4,22,154,...].
Stirling transform of A001563(n+1)=[4,18,96,600,...] is a(n+1)=[4,22,154,...]. (End)

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

A005649(n)=2*a(n), if n>0.
Pairwise sums of A091346.
Cf. A090665.

b:= proc(n, m) option remember;
     `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0)/2:
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 14 2025

a[n_] := (-1)^n (PolyLog[-n - 1, 2] - PolyLog[-n, 2])/8;
Array[a, 21] (* Jean-François Alcover, Sep 10 2018, from A005649 *)

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

E.g.f.: (1/(2-exp(x))^2-1)/2. - Michael Somos, Mar 04 2004
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k+4) - 2*x^2*(k+1)*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n) = Sum_{k=1..n} k * A090665(n,k). - Alois P. Heinz, Feb 20 2025

A302922 Raw moments of a Fibonacci-geometric probability distribution.

Original entry on oeis.org

1, 6, 58, 822, 15514, 366006, 10361818, 342239862, 12918651034, 548600581686, 25885279045978, 1343513774912502, 76071145660848154, 4666162902628259766, 308236822886732856538, 21815861409181135034742, 1646982315540717414270874, 132109620398598537723816246

Offset: 0

Albert Gordon Smith, Apr 15 2018

nonn

If F(k) is the k-th Fibonacci number, where F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
For example, it is the probability that k coin tosses are required to get two heads in a row, or the probability that a random series of k bits has its first two consecutive 1's at the end.
The g.f. for this distribution is g(x) = x^2/(4-2x-x^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The n-th moments about zero of this distribution, known as raw moments, are defined by a(n) = Sum_{k>=1} (k^n)p(k). They appear to be integers and form this sequence.
The e.g.f. for the raw moments is g(e^x) = 1 + 6x + 58x^2/2! + 822x^3/3! + ....
For n >= 1, a(n) appears to be even.
Dividing these terms by 2 gives sequence A302923.
The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.
The central moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302925.
The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), also appear to be integers. They form sequence A302926.
The cumulants also appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.
Note: Another probability distribution on the positive integers that has integral moments and cumulants is the geometric distribution p(k)=1/2^k. The sequences related to these moments are A000629, A000670, A052841, and A091346.

			a(0)=1 is the 0th raw moment of the distribution, which is the total probability.
a(1)=6 is the 1st raw moment, known as the mean of the distribution. It is the arithmetic average of integers following the distribution.
a(2)=58 is the 2nd raw moment. It is the arithmetic average of the squares of integers following the distribution.

Albert Gordon Smith, Table of n, a(n) for n = 0..300
Ignas Gasparavičius, Andrius Grigutis, and Juozas Petkelis, Picturesque convolution-like recurrences and partial sums' generation, arXiv:2507.23619 [math.NT], 2025. See p. 23.
Christopher Genovese, Double Heads

Raw half-moments: A302923.
Central moments: A302924.
Central half-moments: A302925.
Cumulants: A302926.
Half-cumulants: A302927.
Cf. A000629, A000670, A052841, A091346.

Module[{max, r, g},
  max = 17;
  r = Range[0, max];
  g[x_] := x^2/(4 - 2 x - x^2);
  r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x]
]

Vec(serlaplace(exp(2*x)/(4-2*exp(x)-exp(2*x)))) \\ Michel Marcus, Apr 17 2018

In the following,
F(k) is the k-th Fibonacci number, as defined in the Comments.
phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1-sqrt(5))/2.
Li(s,z) is the polylogarithm of order s and argument z.
When s is a negative integer as it is here, Li(s,z) is a rational function of z: Li(-n,z) = (z(d/dz))^n(z/(1-z)).
For n>=0:
a(n) = Sum_{k>=1} ((k^n)(F(k-1)/2^k));
a(n) = Sum_{k>=1} ((k^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));
a(n) = (Li(-n,phi/2)/phi-Li(-n,psi/2)/psi)/sqrt(5).
E.g.f.: g(e^x) where g(x) = x^2/(4-2x-x^2) is the g.f. for the probability distribution.
a(n) ~ n! * (5 - sqrt(5)) / (10 * (log(sqrt(5) - 1))^(n+1)). - Vaclav Kotesovec, Apr 13 2022

A302923 Raw half-moments of a Fibonacci-geometric probability distribution.

Original entry on oeis.org

3, 29, 411, 7757, 183003, 5180909, 171119931, 6459325517, 274300290843, 12942639522989, 671756887456251, 38035572830424077, 2333081451314129883, 154118411443366428269, 10907930704590567517371, 823491157770358707135437, 66054810199299268861908123

Offset: 1

Albert Gordon Smith, Apr 15 2018

nonn

If F(k) is the k-th Fibonacci number, where F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
For example, it is the probability that k coin tosses are required to get two heads in a row, or the probability that a random series of k bits has its first two consecutive 1's at the end.
The g.f. for this distribution is g(x) = x^2/(4-2x-x^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The n-th moments about zero of this distribution, known as raw moments, are defined by a(n) = Sum_{k>=1} (k^n)*p(k). They appear to be integers and form A302922.
The e.g.f. for the raw moments is g(e^x) = 1 + 6x + 58x^2/2! + 822x^3/3! + ....
For n >= 1, the raw moments appear to be even. Dividing them by 2 gives this sequence of raw half-moments.
The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.
The central moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302925.
The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), also appear to be integers. They form sequence A302926.
The cumulants also appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.
Note: Another probability distribution on the positive integers that has integral moments and cumulants is the geometric distribution p(k)=1/2^k. The sequences related to these moments are A000629, A000670, A052841, A091346.

			a(1)=3 is half the first raw moment of the distribution. It is half the arithmetic average of integers following the distribution.
a(2)=29 is half the second raw moment. It is half the arithmetic average of the squares of integers following the distribution.

Albert Gordon Smith, Table of n, a(n) for n = 1..300
Christopher Genovese, Double Heads

Raw moments: A302922.
Central moments: A302924.
Central half-moments: A302925.
Cumulants: A302926.
Half-cumulants: A302927.
Cf. A000629, A000670, A052841, A091346.

Module[{max, r, g, rawMoments},
  max = 17;
  r = Range[0, max];
  g[x_] := x^2/(4 - 2 x - x^2);
  rawMoments = r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x];
  Rest[rawMoments]/2
]

In the following,
F(k) is the k-th Fibonacci number, as defined in the Comments.
phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1-sqrt(5))/2.
Li(s,z) is the polylogarithm of order s and argument z.
When s is a negative integer as it is here, Li(s,z) is a rational function of z: Li(-n,z) = (z(d/dz))^n(z/(1-z)).
For n>=1:
a(n) = (1/2)A302922(n);
a(n) = (1/2)Sum_{k>=1} ((k^n)(F(k-1)/2^k));
a(n) = (1/2)Sum_{k>=1} ((k^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));
a(n) = (1/2)(Li(-n,phi/2)/phi-Li(-n,psi/2)/psi)/sqrt(5).
E.g.f.: (1/2)g(e^x) where g(x) = x^2/(4-2x-x^2) is the g.f. for the probability distribution.

A302924 Central moments of a Fibonacci-geometric probability distribution.

Original entry on oeis.org

1, 0, 22, 210, 4426, 102330, 2906362, 95952570, 3622138906, 153816150810, 7257695358202, 376693381614330, 21328770664314586, 1308295248437904090, 86423208789970618042, 6116714829331037666490, 461779664078480243085466, 37040796099362864616022170

Offset: 0

Albert Gordon Smith, Apr 15 2018

nonn

If F(k) is the k-th Fibonacci number, where F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
For example, it is the probability that k coin tosses are required to get two heads in a row, or the probability that a random series of k bits has its first two consecutive 1's at the end.
The g.f. for this distribution is g(x) = x^2/(4-2x-x^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The mean of this distribution is 6. (See A302922.)
The n-th moments about the mean, known as central moments, are defined by a(n) = Sum_{k>=1} ((k-6)^n)p(k). They appear to be integers and form this sequence.
For n >= 1, a(n) appears to be even. Dividing these terms by 2 gives sequence A302925.
The raw moments (i.e., the moments about zero) also appear to be integers. This is sequence A302922.
The raw moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302923.
The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), also appear to be integers. They form sequence A302926.
The cumulants also appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.
Note: Another probability distribution on the positive integers that has integral moments and cumulants is the geometric distribution p(k)=1/2^k. The sequences related to these moments are A000629, A000670, A052841, and A091346.

			a(0)=1 is the 0th central moment of the distribution, which is the total probability.
a(1)=0 is the 1st central moment, or the "mean about the mean". It is zero by definition of central moments.
a(2)=22 is the 2nd central moment, known as the variance or the square of the standard deviation. It measures how far integers following the distribution are from the mean by averaging the squares of their differences from the mean.

Albert Gordon Smith, Table of n, a(n) for n = 0..300
Christopher Genovese, Double Heads

Central half-moments: A302925.
Raw moments: A302922.
Raw half-moments: A302923.
Cumulants: A302926.
Half-cumulants: A302927.
Cf. A000629, A000670, A052841, A091346.

Module[{max, r, g, moments},
  max = 17;
  r = Range[0, max];
  g[x_] := x^2/(4 - 2 x - x^2);
  moments = r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x];
  Table[Sum[Binomial[n, k] moments[[k + 1]] (-6)^(n - k), {k, 0, n}], {n, 0, max}]
]

In the following,
F(k) is the k-th Fibonacci number, as defined in the Comments.
phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1-sqrt(5))/2.
LerchPhi(z,s,a) = Sum_{k>=0} z^k/(a+k)^s is the Lerch transcendant.
For n >= 0:
a(n) = Sum_{k>=1} (((k-6)^n)(F(k-1)/2^k));
a(n) = Sum_{k>=1} (((k-6)^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));
a(n) = (LerchPhi(phi/2,-n,-5)-LerchPhi(psi/2,-n,-5))/(2 sqrt(5));
a(n) = Sum_{k=0..n} (binomial(n,k)*A302922(k)*(-6)^(n-k)).

A302925 Central half-moments of a Fibonacci-geometric probability distribution.

Original entry on oeis.org

0, 11, 105, 2213, 51165, 1453181, 47976285, 1811069453, 76908075405, 3628847679101, 188346690807165, 10664385332157293, 654147624218952045, 43211604394985309021, 3058357414665518833245, 230889832039240121542733, 18520398049681432308011085

Offset: 1

Albert Gordon Smith, Apr 15 2018

nonn

If F(k) is the k-th Fibonacci number, where F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
For example, it is the probability that k coin tosses are required to get two heads in a row, or the probability that a random series of k bits has its first two consecutive 1's at the end.
The g.f. for this distribution is g(x) = x^2/(4-2x-x^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The mean of this distribution is 6. (See A302922.)
The n-th moments about the mean, known as central moments, are defined by a(n) = Sum_{k>=1} ((k-6)^n)p(k). They appear to be integers and form A302924.
For n >= 1, that sequence appears to be even. Dividing those terms by 2 gives this sequence.
The raw moments (i.e., the moments about zero) also appear to be integers. This is sequence A302922.
The raw moments also appear to be even for n >= 1. Dividing them by 2 gives sequence A302923.
The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), also appear to be integers. They form sequence A302926.
The cumulants also appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.
Note: Another probability distribution on the positive integers that has integral moments is the geometric distribution p(k)=1/2^k. The sequences related to these moments are A000629, A000670, A052841, and A091346.

			a(1)=0 is half the 1st central moment of the distribution, or half the "mean about the mean". It is zero by definition of central moments.
a(2)=11 is half the 2nd central moment, or half the variance, or half the square of the standard deviation.

Albert Gordon Smith, Table of n, a(n) for n = 1..300
Christopher Genovese, Double Heads

Central moments: A302924.
Raw moments: A302922.
Raw half-moments: A302923.
Cumulants: A302926.
Half-cumulants: A302927.
Cf. A000629, A000670, A052841, A091346.

Module[{max, r, g, moments, centralMoments},
  max = 17;
  r = Range[0, max];
  g[x_] := x^2/(4 - 2 x - x^2);
  moments = r! CoefficientList[Normal[Series[g[Exp[x]], {x, 0, max}]], x];
  centralMoments = Table[Sum[Binomial[n, k] moments[[k + 1]] (-6)^(n - k), {k, 0, n}], {n, 0, max}];
  Rest[centralMoments]/2
]

In the following,
F(k) is the k-th Fibonacci number, as defined in the Comments.
phi=(1+sqrt(5))/2 is the golden ratio, and psi=(1-sqrt(5))/2.
LerchPhi(z,s,a) = Sum_{k>=0} z^k/(a+k)^s is the Lerch transcendant.
For n>=1:
a(n) = (1/2)A302924(n);
a(n) = (1/2)Sum_{k>=1} (((k-6)^n)(F(k-1)/2^k));
a(n) = (1/2)Sum_{k>=1} (((k-6)^n)(((phi^(k-1)-psi^(k-1))/sqrt(5))/2^k));
a(n) = (1/2)(LerchPhi(phi/2,-n,-5)-LerchPhi(psi/2,-n,-5))/(2 sqrt(5));
a(n) = (1/2)Sum_{k=0..n} (binomial(n,k)A302922(k)(-6)^(n-k)).

A302926 Cumulants of a Fibonacci-geometric probability distribution.

Original entry on oeis.org

0, 6, 22, 210, 2974, 56130, 1324222, 37489410, 1238235454, 46740118530, 1984855550782, 93653819396610, 4860878501987134, 275227990564092930, 16882335978752910142, 1115211301788480951810, 78930528072274523870014, 5958837996496319756259330

Offset: 0

Albert Gordon Smith, Apr 15 2018

nonn

If F(k) is the k-th Fibonacci number, where F(0)=0, F(1)=1, and F(n)=F(n-1)+F(n-2), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
For example, it is the probability that k coin tosses are required to get two heads in a row, or the probability that a random series of k bits has its first two consecutive 1's at the end.
The g.f. for this distribution is g(x) = x^2/(4-2x-x^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), appear to be integers and form this sequence.
The cumulants appear to be even for n >= 0. Dividing them by 2 gives sequence A302927.
The n-th moments about zero of this distribution, known as raw moments, are defined by a(n) = Sum_{k>=1} (k^n)p(k). They also appear to be integers and form sequence A302922.
For n >= 1, the raw moments also appear to be even. Dividing them by 2 gives sequence A302923.
The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.
For n >= 1, the central moments also appear to be even. Dividing them by 2 gives sequence A302925.
Note: Another probability distribution on the positive integers that has integral moments and cumulants is the geometric distribution p(k)=1/2^k. The sequences related to these moments are A000629, A000670, A052841, and A091346.
Variant of A103437. - R. J. Mathar, Jun 09 2018

			a(0)=0 is the 0th cumulant of the distribution. The 0th cumulant is always zero.
a(1)=6 is the 1st cumulant, which is always the mean.
a(2)=22 is the 2nd cumulant, which is always the variance.

Albert Gordon Smith, Table of n, a(n) for n = 0..300
Christopher Genovese, Double Heads

Half-cumulants: A302927.
Raw moments: A302922.
Raw half-moments: A302923.
Central moments: A302924.
Central half-moments: A302925.
Cf. A000629, A000670, A052841, A091346, A103437.

Module[{max, r, g},
  max = 17;
  r = Range[0, max];
  g[x_] := x^2/(4 - 2 x - x^2);
  r! CoefficientList[Normal[Series[Log[g[Exp[x]]], {x, 0, max}]], x]
]

E.g.f.: log(g(e^x)) where g(x) = x^2/(4-2x-x^2) is the g.f. for the probability distribution.

A302927 Half-cumulants of a Fibonacci-geometric probability distribution.

Original entry on oeis.org

0, 3, 11, 105, 1487, 28065, 662111, 18744705, 619117727, 23370059265, 992427775391, 46826909698305, 2430439250993567, 137613995282046465, 8441167989376455071, 557605650894240475905, 39465264036137261935007, 2979418998248159878129665

Offset: 0

Albert Gordon Smith, Apr 15 2018

nonn

If F(k) is the k-th Fibonacci number A000045(k), then p(k)=F(k-1)/2^k is a normalized probability distribution on the positive integers.
For example, it is the probability that k coin tosses are required to get two heads in a row, or the probability that a random series of k bits has its first two consecutive 1's at the end.
The g.f. for this distribution is g(x) = x^2/(4-2x-x^2) = (1/4)x^2 + (1/8)x^3 + (1/8)x^4 + (3/32)x^5 + ....
The cumulants of this distribution, defined by the cumulant e.g.f. log(g(e^x)), appear to be integers. They form sequence A302926.
The cumulants appear to be even for n >= 0. Dividing them by 2 gives this sequence.
The n-th moments about zero of this distribution, known as raw moments, are defined by a(n) = Sum_{k>=1} (k^n)p(k). They also appear to be integers and form sequence A302922.
For n >= 1, the raw moments also appear to be even. Dividing them by 2 gives sequence A302923.
The central moments (i.e., the moments about the mean) also appear to be integers. They form sequence A302924.
For n >= 1, the central moments appear to be even. Dividing them by 2 gives sequence A302925.
Note: Another probability distribution on the positive integers that has integral moments and cumulants is the geometric distribution p(k)=1/2^k. The sequences related to these moments are A000629, A000670, A052841, and A091346.

			a(0)=0 is half the 0th cumulant of the distribution. The 0th cumulant is always zero.
a(1)=3 is half the 1st cumulant, which is half the mean.
a(2)=11 is half the 2nd cumulant, which is half the variance.

Albert Gordon Smith, Table of n, a(n) for n = 0..300
Christopher Genovese, Double Heads

Cumulants: A302926.
Raw moments: A302922.
Raw half-moments: A302923.
Central moments: A302924.
Central half-moments: A302925.
Cf. A000045, A000629, A000670, A052841, A091346.

Module[{max, r, g},
  max = 17;
  r = Range[0, max];
  g[x_] := x^2/(4 - 2 x - x^2);
  (1/2) r! CoefficientList[Normal[Series[Log[g[Exp[x]]], {x, 0, max}]], x]
]

concat(0, Vec(serlaplace(log(exp(2*x)/(4-2*exp(x)-exp(2*x))))/2)) \\ Michel Marcus, Apr 17 2018

E.g.f.: (1/2)*log(g(e^x)) where g(x) = x^2/(4-2*x-x^2) is the g.f. for the probability distribution.

A032109 "BIJ" (reversible, indistinct, labeled) transform of 1,1,1,1,...

Original entry on oeis.org

1, 1, 2, 7, 38, 271, 2342, 23647, 272918, 3543631, 51123782, 811316287, 14045783798, 263429174191, 5320671485222, 115141595488927, 2657827340990678, 65185383514567951, 1692767331628422662, 46400793659664205567, 1338843898122192101558, 40562412499252036940911

Offset: 0

Christian G. Bower

nonn

Starting (1, 2, 7, 38, 271, ...) = A008292 * [1, 1, 2, 4, 8, ...]. - Gary W. Adamson, Dec 25 2008
The inverse binomial transform is 1, 0, 1, 3, 19, 135, 1171, 11823, 136459, ..., see A091346. - R. J. Mathar, Oct 17 2012
Stirling transform of A001710. - Anton Zakharov, Aug 06 2016
For n even (resp. n odd), a(n) counts the ordered partitions of {1,2,...,n} with an even (resp. odd) number of blocks, and a(n)-1 counts the ordered partitions of {1,2,...,n} with an odd (resp. even) number of blocks. - Jose A. Rodriguez, Feb 04 2021

Alois P. Heinz, Table of n, a(n) for n = 0..424 (first 101 terms from R. J. Mathar)
C. G. Bower, Transforms (2)

A000629, A000670, A002050, A032109, A052856, A076726 are all more-or-less the same sequence. - N. J. A. Sloane, Jul 04 2012
A052856(n)=2*a(n), if n>0.
Cf. A008292, A001710.

A032109 := proc(n)
    (A000670(n)+1)/2 ;
end proc: # R. J. Mathar, Oct 17 2012
a := n -> (polylog(-n, 1/2)+`if`(n=0,3,2))/4:
seq(round(evalf(a(n), 32)), n=0..18); # Peter Luschny, Nov 03 2015
# alternative Maple program:
b:= proc(n, m) option remember; `if`(n=0, m!,
      add(b(n-1, max(m, j)), j=1..m+1))
    end:
a:= n-> (b(n,0)+1)/2:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

Table[(PolyLog[-n, 1/2] + 2 + KroneckerDelta[n])/4, {n, 0, 20}] (* Vladimir Reshetnikov, Nov 02 2015 *)

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

list(n)=my(v=Vec(subst((1-y^2/2)/(1-y),y,exp(x+x*O(x^n))-1)));vector(n+1,i,v[i]*(i-1)!) \\ Charles R Greathouse IV, Oct 17 2012

E.g.f.: (e^(2*x)-2*e^x-1)/(2*e^x-4).
a(n) = (A000670(n)+1)/2. - Vladeta Jovovic, Apr 13 2003
a(n) = A052875(n)/2 + 1. - Max Alekseyev, Jan 31 2021
a(n) ~ sqrt(Pi/2)*n^(n+1/2)/(2*log(2)^(n+1)*exp(n)). - Ilya Gutkovskiy, Aug 06 2016
a(n) = Sum_{s in S_n^even} Product_{i=1..n} binomial(i,s(i)-1), where s ranges over the set S_n^even of even permutations of [n]. - Jose A. Rodriguez, Feb 02 2021

A308804 Triangular table of coefficients of p in p^(k+2)/(1-p) LerchPhi(1-p,-1-k,(p-1)/p) as function of k=1..n.

Original entry on oeis.org

1, 2, -1, 9, -9, 1, 44, -66, 24, -1, 265, -530, 320, -55, 1, 1854, -4635, 3940, -1275, 118, -1, 14833, -44499, 48825, -23485, 4571, -245, 1, 133496, -467236, 628544, -403270, 123368, -15400, 500, -1, 1334961, -5339844, 8510376, -6841674, 2885694, -598416, 49914, -1011, 1, 14684570, -66080565, 121759560, -117782490, 63630588, -18808230, 2752320, -157785, 2034, -1

Offset: 1

Wouter Meeussen, Jun 25 2019

Relations to other sequences are tentative (checked up to 24 rows). Related to the central moments of a geometric probability distribution.

Each row sums to 1. Row sums of absolute values are A091346. First column is A000166. First moments appear to be A052515 with shifted index.

Wolfram Language & System Documentation Center, LerchPhi (then click in Applications).

Cf. A091346, A000166, A052515.

Table[CoefficientList[ p^(k + 2)/(1 - p) LerchPhi[1 - p, -k - 1, (-1 + p)/p], p], {k, 1, 12}]

cp's OEIS Frontend

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