A103446 -id:A103446 - cp's OEIS frontend

A218482 First differences of the binomial transform of the partition numbers (A000041).

1, 1, 3, 8, 21, 54, 137, 344, 856, 2113, 5179, 12614, 30548, 73595, 176455, 421215, 1001388, 2371678, 5597245, 13166069, 30873728, 72185937, 168313391, 391428622, 908058205, 2101629502, 4853215947, 11183551059, 25718677187, 59030344851, 135237134812, 309274516740

Offset: 0

Paul D. Hanna, Oct 29 2012

nonn

a(n) = A103446(n) for n>=1; here a(0) is set to 1 in accordance with the definition and other important generating functions.
From Gus Wiseman, Dec 12 2022: (Start)
Also the number of sequences of compositions (A133494) with weakly decreasing lengths and total sum n. For example, the a(0) = 1 through a(3) = 8 sequences are:
  ()  ((1))  ((2))     ((3))
             ((11))    ((12))
             ((1)(1))  ((21))
                       ((111))
                       ((1)(2))
                       ((2)(1))
                       ((11)(1))
                       ((1)(1)(1))
The case of constant lengths is A101509.
The case of strictly decreasing lengths is A129519.
The case of sequences of partitions is A141199.
The case of twice-partitions is A358831.
(End)

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...
The g.f. equals the product:
A(x) = (1-x)/((1-x)-x) * (1-x)^2/((1-x)^2-x^2) * (1-x)^3/((1-x)^3-x^3) * (1-x)^4/((1-x)^4-x^4) * (1-x)^5/((1-x)^5-x^5) * (1-x)^6/((1-x)^6-x^6) * (1-x)^7/((1-x)^7-x^7) *...
and also equals the series:
A(x) = 1  +  x*(1-x)/((1-x)-x)^2  +  x^4*(1-x)^2/(((1-x)-x)*((1-x)^2-x^2))^2  +  x^9*(1-x)^3/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3))^2  +  x^16*(1-x)^4/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3)*((1-x)^4-x^4))^2 +...

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A000219, A011782, A055887, A063834, A075900, A098407, A101509, A103446, A129519, A141199, A218481.

b:= proc(n) option remember;
      add(combinat[numbpart](k)*binomial(n,k), k=0..n)
    end:
a:= n-> b(n)-b(n-1):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 19 2014

Flatten[{1, Table[Sum[Binomial[n-1,k]*PartitionsP[k+1],{k,0,n-1}],{n,1,30}]}] (* Vaclav Kotesovec, Jun 25 2015 *)

{a(n)=sum(k=0,n,(binomial(n,k)-if(n>0,binomial(n-1,k)))*numbpart(k))}
for(n=0,40,print1(a(n),", "))

{a(n)=local(X=x+x*O(x^n));polcoeff(prod(k=1,n,(1-x)^k/((1-x)^k-X^k)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(sum(m=0,n,x^m*(1-x)^(m*(m-1)/2)/prod(k=1,m,((1-x)^k - X^k))),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(sum(m=0,n,x^(m^2)*(1-X)^m/prod(k=1,m,((1-x)^k - x^k)^2)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(exp(sum(m=1,n+1,x^m/((1-x)^m-X^m)/m)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(exp(sum(m=1,n+1,sigma(m)*x^m/(1-X)^m/m)),n)}

{a(n)=local(X=x+x*O(x^n));polcoeff(prod(k=1,n,(1 + x^k/(1-X)^k)^valuation(2*k,2)),n)}

G.f.: Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).
G.f.: Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k).
G.f.: Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.
G.f.: exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).
G.f.: exp( Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n ), where sigma(n) is the sum of divisors of n (A000203).
G.f.: Product_{n>=1} (1 + x^n/(1-x)^n)^A001511(n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-2) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

A025168 Expansion of e.g.f.: exp(x/(1-2*x)).

Original entry on oeis.org

1, 1, 5, 37, 361, 4361, 62701, 1044205, 19748177, 417787921, 9770678101, 250194150581, 6959638411705, 208919770666777, 6729933476435261, 231512615111396221, 8469125401589550241, 328241040596380393505, 13434223364220816489637, 578931271898150002093381

Offset: 0

Wouter Meeussen

From Peter Bala, Nov 21 2017: (Start)
The sequence terms have the form 4*m + 1 (follows from the recurrence).
For k = 2,3,4,... the difference a(n+k) - a(n) is divisible by k (proof by induction on n making use of the recurrence - the starting case a(k) == a(0) (mod k) for all k follows from the sum formula for a(k)). Hence for each k, the sequence b(n) == a(n) (mod k) is periodic with the exact period dividing k. (End)
Compound Poisson distribution with parameter 1 and distribution Geometric(1/2) has a probability mass function p_n = a(n)*e^(-1/2)/(4^n*n!). More specifically, let S = Sum_{i=0..N} X_i where X_i's are i.i.d. random variables with Geometric(1/2) distribution (i.e., Pr{X_i = k} = 1/2^(k+1) for k=0,1,2...) and N is a random variable with Poisson(1) distribution independent of all X_i's. Then Pr{S=n} = a(n)*e^(-1/2)/(4^n*n!) = a(n)*e^(-1/2)/A047053(n) for nonnegative integers n. - Xiaohan Zhang, Nov 16 2022

Robert Israel, Table of n, a(n) for n = 0..400
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, arXiv:quant-ph/0312202, 2003; J. Phys. A 37 (2004), 3475-3487.
N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
Thomas Wieder, Expanded definitions of A103446 and A025168

Cf. A000012, A000262, A103446.

with(combstruct); SetSeqSeqL := [T, {T=Set(S), S=Sequence(U,card >= 1), U=Sequence(Z,card >=1)},labeled];
f:= gfun:-rectoproc({a(n) = (4*n-3)*a(n-1) - 4*(n-2)*(n-1)*a(n-2),a(0)=1,a(1)=1},a(n),remember):
map(f, [$0..30]); # Robert Israel, Nov 21 2017

Table[ n! 2^n LaguerreL[ n, 1, -1/2 ], {n, 0, 12} ]
With[{nn=20},CoefficientList[Series[Exp[x/(1-2x)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2012 *)

A025168 = lambda n: hypergeometric([-n,-n+1], [], 2)
[Integer(A025168(n).n(100)) for n in range(20)] # Peter Luschny, Sep 22 2014

Second LAH transform of A000012. LAH transform of A000262. a(n) = Sum_{k=0..n} 2^(n-k)*n!/k!*binomial(n-1, k-1). - Vladeta Jovovic, Oct 17 2003
Define f_1(x), f_2(x), ... such that f_1(x) = e^x, f_{n+1}(x) = (d/dx)(x^2*f_n(x)), for n=2,3,.... Then a(n) = e^(-1/2)*4*(n-1)*f_n(1/2). - Milan Janjic, May 30 2008
From Vaclav Kotesovec, Jun 22 2013: (Start)
D-finite with recurrence: a(n) = (4*n-3)*a(n-1) - 4*(n-2)*(n-1)*a(n-2).
a(n) ~ 2^(n-3/4)*n^(n-1/4)*exp(sqrt(2*n)-n-1/4) * (1-1/(3*sqrt(2*n))).
(End)
E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - x/(x + (k+1)*(1-2*x)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 09 2013
a(n) = hypergeometric([-n,-n+1],[],2). - Peter Luschny, Sep 22 2014
Sum_{n>=0} a(n)/(4^n*n!) = sqrt(e) = A019774. -Xiaohan Zhang, Nov 16 2022

Corrected and extended by Vladeta Jovovic, Sep 08 2002

A320590 Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)^k).

Original entry on oeis.org

1, 1, 1, 0, 1, -2, 5, -12, 28, -63, 137, -290, 604, -1253, 2617, -5537, 11870, -25666, 55617, -120103, 257582, -548119, 1158437, -2437114, 5117165, -10748530, 22621055, -47728657, 100932549, -213750621, 452855190, -958925784, 2028187595, -4283531490, 9033779224

Offset: 0

Ilya Gutkovskiy, Oct 16 2018

sign

The zero-based binomial transform of this sequence is A000070, and if we remove first terms it becomes A000041.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000203, A103446, A218482, A320568.
Row n=1 of A175804 (except first term). Row n=0 is A281425.
The version for strict partitions is A320591, row n=1 of A378622, first column A293467.
A000009 counts strict integer partitions, differences A087897, A378972.
A000041 counts integer partitions, differences A002865.
Cf. A047966, A008284, A377056, A378621.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1 - x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

seq(coeff(series(mul(1/(1-x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 16 2018

nmax = 34; CoefficientList[Series[Product[1/(1 - x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 34; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

m=50; x='x+O('x^m); Vec(prod(k=1, m+2, 1/(1 - x^k/(1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

G.f.: exp(Sum_{k>=1} x^k/(k*((1 + x)^k - x^k))).

G.f.: exp(Sum_{k>=1} sigma(k)*x^k/(k*(1 + x)^k)).

A185003 a(n) = Sum_{k=1..n} binomial(n,k)*sigma(k).

Original entry on oeis.org

1, 5, 16, 45, 116, 284, 673, 1557, 3535, 7910, 17502, 38376, 83500, 180479, 387881, 829605, 1766998, 3749765, 7931114, 16724870, 35173777, 73794660, 154485527, 322771344, 673155141, 1401536934, 2913490375, 6047714599, 12536770558, 25956242579, 53678385266

Offset: 1

Paul D. Hanna, Feb 04 2012

nonn

			L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 45*x^4/4 + 116*x^5/5 + 284*x^6/6 +...
where exponentiation yields A103446 (with offset=0):
exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A103446, A222115, A000203, A160399.

[&+[Binomial(n,k)*DivisorSigma(1,k):k in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019

[&+[&+[i*Binomial(n,i*j):j in [1..n]]:i in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019

with(numtheory): seq(add(binomial(n,i)*sigma(i), i=1..n), n=1..40); # Ridouane Oudra, Nov 12 2019

Table[Sum[Binomial[n, k] DivisorSigma[1, k], {k, n}], {n,50}] (* G. C. Greubel, Jun 03 2017 *)

{a(n)=sum(k=1,n,sigma(k)*binomial(n,k))}
for(n=1,30,print1(a(n),", "))

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(m=1, n+1, x^m/((1-x)^m-X^m)/m), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(k=1, n, k*log(1-X)-log((1-x)^k-X^k)), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), n)}

Logarithmic derivative of A103446 (with offset=0), which describes the binomial transform of partitions.
From Paul D. Hanna, Jun 01 2013: (Start)
L.g.f.: Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
L.g.f.: Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
L.g.f.: Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
L.g.f.: Sum_{n>=1} A001511(n) * log(1 + x^n/(1-x)^n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
a(n) = A222115(n) - 1. (End)
a(n) ~ Pi^2/12 * n * 2^n. - Vaclav Kotesovec, Dec 30 2015
a(n) = Sum_{i=1..n} Sum_{j=1..n} i*binomial(n,i*j). - Ridouane Oudra, Nov 12 2019

A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

Original entry on oeis.org

1, 1, 0, 1, -2, 4, -7, 11, -16, 23, -36, 65, -129, 256, -473, 772, -1028, 835, 776, -5755, 17562, -41750, 86678, -165145, 299949, -541837, 1020029, -2068203, 4509512, -10252952, 23465297, -52762788, 115160832, -243018459, 496094524, -982431070, 1894710043, -3574095362

Offset: 0

Ilya Gutkovskiy, Oct 16 2018

After the first term, this is the second term of the n-th differences of A000009, or column n=1 of A378622. - Gus Wiseman, Feb 03 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000593, A320589, A307548.
The version for non-strict partitions is A320590, row n=1 of A175804.
Column n=1 (except first term) of A378622. See also A293467, A377285, A378970, A378971, A380412 (column n=0).
A000009 counts strict integer partitions, differences A087897, A378972.
A266232 gives zero-based binomial transform of strict partitions, differences A129519.
Cf. A000041, A007442, A002865, A008284, A095195, A103446, A116608, A218482, A281425, A320568.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1 + x^k/(1 + x)^k): k in [1..(m+2)]]) )); // G. C. Greubel, Oct 29 2018

seq(coeff(series(mul((1+x^k/(1+x)^k),k=1..n),x,n+1), x, n), n = 0 .. 37); # Muniru A Asiru, Oct 16 2018

nmax = 37; CoefficientList[Series[Product[(1 + x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d, {d, Divisors[k]}] x^k/(k (1 + x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]
Prepend[Table[Differences[PartitionsQ/@Range[0,k+1],k][[2]],{k,0,30}],1] (* Gus Wiseman, Jan 29 2025 *)

m=50; x='x+O('x^m); Vec(prod(k=1, m+2, (1 + x^k/(1 + x)^k))) \\ G. C. Greubel, Oct 29 2018

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*((1 + x)^k - x^k))).
G.f.: exp(Sum_{k>=1} A000593(k)*x^k/(k*(1 + x)^k)).
From Peter Bala, Dec 22 2020: (Start)
O.g.f.: Sum_{n >= 0}  x^(n*(n+1)/2)/Product_{k = 1..n} ((1 + x)^k - x^k). Cf. A307548.
Conjectural o.g.f.: (1/2) * Sum_{n >= 0} x^(n*(n-1)/2)*(1 + x)^n/( Product_{k = 1..n} ( (1 + x)^k - x^k ) ). (End)
a(n+1) = Sum_{k=0..n} (-1)^(n-k) binomial(n,k) A000009(k+1). - Gus Wiseman, Feb 03 2025

A320563 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^k.

Original entry on oeis.org

1, 1, 4, 13, 41, 125, 374, 1103, 3213, 9259, 26430, 74806, 210095, 585890, 1623240, 4470232, 12241799, 33349751, 90410255, 243977941, 655553258, 1754265279, 4676358086, 12420299846, 32873598566, 86721264126, 228051843891, 597905347237, 1563071037798, 4074973824099

Offset: 0

Ilya Gutkovskiy, Oct 15 2018

nonn

First differences of the binomial transform of A000219.

N. J. A. Sloane, Transforms

Cf. A000219, A001157, A103446, A218482, A294500, A320564.

seq(coeff(series(mul((1-x^k/(1-x)^k)^(-k),k=1..n),x,n+1), x, n), n = 0 .. 29); # Muniru A Asiru, Oct 15 2018

nmax = 29; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)).

a(n) ~ Zeta(3)^(7/36) * 2^(n - 11/18) * exp(3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) + (1 - Zeta(3))/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 15 2018

A307310 Expansion of Product_{k>=1} (1 - x^k/(1 - x)^k).

Original entry on oeis.org

1, -1, -2, -3, -4, -4, -1, 9, 34, 89, 200, 409, 779, 1394, 2339, 3624, 4974, 5323, 1682, -13279, -56222, -163136, -408768, -943275, -2059237, -4310179, -8712425, -17072901, -32486302, -60006278, -107341413, -184979170, -303998680, -467127625, -642495990, -696247140

Offset: 0

Ilya Gutkovskiy, Apr 02 2019

sign

First differences of the binomial transform of A010815.

Convolution inverse of A218482.

Cf. A010815, A103446, A129519, A218482, A307311.

a:=series(mul((1-x^k/(1-x)^k),k=1..100),x=0,35): seq(coeff(a,x,n),n=0..34); # Paolo P. Lava, Apr 02 2019

nmax = 35; CoefficientList[Series[Product[(1 - x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

A137151 Triangle read by rows: A007318 * A026794.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 13, 4, 3, 1, 34, 9, 6, 4, 1, 88, 22, 11, 10, 5, 1, 225, 55, 22, 20, 15, 6, 1, 569, 137, 50, 36, 35, 21, 7, 1, 1425, 338, 122, 65, 70, 56, 28, 8, 1, 3538, 826, 302, 130, 127, 126, 84, 36, 9, 1, 8717, 2002, 740, 296, 221, 252, 210, 120, 45, 10, 1

Offset: 1

Gary W. Adamson, Jan 23 2008

Row sums = A103446: (1, 3, 8, 21, 54, 137, 344, ...).

			First few rows of the triangle:
    1;
    2,   1;
    5,   2,  1;
   13,   4,  3,  1;
   34,   9,  6,  4,  1;
   88,  22, 11, 10,  5,  1;
  225,  55, 22, 20, 15,  6, 1;
  569, 137, 50, 36, 35, 21, 7, 1;
  ...

Cf. A026794, A103446.

Binomial transform of the partition triangle, A026794.

A307679 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^(1/k).

Original entry on oeis.org

1, 1, 5, 35, 323, 3679, 49819, 781465, 13923545, 277563617, 6118251461, 147715469131, 3875706370315, 109781717161375, 3338229675519803, 108443658227589329, 3747688533281296049, 137273241169036231105, 5311844045472206624005, 216505267421266611639667, 9270689769095765333645651

Offset: 0

Ilya Gutkovskiy, Apr 21 2019

nonn

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 35*x^3/3! + 323*x^4/4! + 3679*x^5/5! + 49819*x^6/6! + 781465*x^7/7! + 13923545*x^8/8! + ...
log(A(x)) = x + 4*x^2/2 + 11*x^3/3 + 27*x^4/4 + 62*x^5/5 + 137*x^6/6 + 296*x^7/7 + 630*x^8/8 + 1326*x^9/9 + ... + A160399(k)*x^k/k + ...

Cf. A000005, A028342, A103446, A160399, A218482, A307680, A320563.

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^(1/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: exp(Sum_{k>=1} d(k)*x^k/(k*(1 - x)^k)), where d(k) is the number of divisors of k (A000005).

a(n) = Sum_{k=0..n} binomial(n-1,k-1)*A028342(k)*n!/k!.

cp's OEIS Frontend

A218482 First differences of the binomial transform of the partition numbers (A000041).

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A025168 Expansion of e.g.f.: exp(x/(1-2*x)).

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A320590 Expansion of Product_{k>=1} 1/(1 - x^k/(1 + x)^k).

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A185003 a(n) = Sum_{k=1..n} binomial(n,k)*sigma(k).

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PARI

PARI

Formula

A320591 Expansion of Product_{k>=1} (1 + x^k/(1 + x)^k).

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PARI

Formula

A320563 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^k.

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