A279215 -id:A279215 - cp's OEIS frontend

A279216 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

1, 1, 7, 25, 86, 269, 862, 2606, 7812, 22704, 64989, 182356, 504414, 1373694, 3693367, 9804435, 25733084, 66808578, 171719539, 437183839, 1103143657, 2760037810, 6850400668, 16873338215, 41260373472, 100196920196, 241712863504, 579416535973, 1380517695672, 3270075208145, 7702580246941

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the pentagonal pyramidal numbers (A002411).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000335, A002411, A279215, A279217, A279218, A279219.

nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k^2 (k + 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

a(n) ~ exp(-Zeta(3)/(8*Pi^2) - Pi^16/(83980800000*Zeta(5)^3) + Zeta'(-3)/2 + (Pi^12/(97200000*2^(2/5)*3^(1/5)*Zeta(5)^(11/5))) * n^(1/5) + (-Pi^8/(108000*2^(4/5)*3^(2/5)*Zeta(5)^(7/5))) * n^(2/5) + (Pi^4/(180*2^(1/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(8/5))) * n^(4/5)) * (3*Zeta(5))^(119/1200) / (2^(181/600) * sqrt(5*Pi) * n^(719/1200)). - Vaclav Kotesovec, Dec 08 2016

A279217 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(4*k-1)/6).

Original entry on oeis.org

1, 1, 8, 30, 108, 357, 1205, 3838, 12083, 36896, 110828, 326281, 946086, 2700026, 7602642, 21128513, 58028309, 157588912, 423534324, 1127102360, 2971764946, 7766890826, 20131080168, 51766513279, 132117237595, 334770353022, 842462217948, 2106183375971, 5232414548275, 12920429411759, 31719180847831

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the hexagonal pyramidal numbers (A002412).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Hexagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000335, A002412, A279215, A279216, A279218, A279219.

nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1)(4 k - 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(4*k-1)/6).

a(n) ~ exp(-Zeta'(-1)/6 - Zeta(3)/(8*Pi^2) - Pi^16/(199065600000*Zeta(5)^3) - Pi^8*Zeta(3)/(6912000*Zeta(5)^2) - Zeta(3)^2/(1440*Zeta(5)) + 2*Zeta'(-3)/3 + (Pi^12/(172800000*2^(4/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(7200*2^(4/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(288000*2^(3/5)*Zeta(5)^(7/5)) - Zeta(3)/(12*2^(3/5)*Zeta(5)^(2/5))) * n^(2/5) + (Pi^4/(360*2^(2/5)*Zeta(5)^(3/5))) * n^(3/5) + 5*(Zeta(5)/2)^(1/5)/2 * n^(4/5)) * Zeta(5)^(173/1800) / (2^(26/225) * sqrt(5*Pi) * n^(1073/1800)). - Vaclav Kotesovec, Dec 08 2016

A279218 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(5*k-2)/6).

Original entry on oeis.org

1, 1, 9, 35, 131, 454, 1601, 5325, 17467, 55588, 173858, 532809, 1607056, 4769263, 13957660, 40302923, 114962909, 324157109, 904247056, 2496917319, 6829241131, 18510038697, 49741367504, 132582175873, 350655140642, 920568519505, 2399692063845, 6213105691838, 15982216140168, 40855658807127, 103814659491641

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the heptagonal pyramidal numbers (A002413).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Heptagonal Pyramidal Number
Index to sequences related to pyramidal numbers

Cf. A000335, A002413, A279215, A279216, A279217, A279219.

nmax=30; CoefficientList[Series[Product[1/(1 - x^k)^(k (k + 1) (5 k - 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(5*k-2)/6).

a(n) ~ exp(-Zeta'(-1)/3 - Zeta(3)/(8*Pi^2) - Pi^16/(388800000000*Zeta(5)^3) - Pi^8*Zeta(3)/(5400000*Zeta(5)^2) - Zeta(3)^2/(450*Zeta(5)) + 5*Zeta'(-3)/6 + (Pi^12/(270000000*2^(2/5)*5^(1/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(4500*2^(2/5) * 5^(1/5)*Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(180000*2^(4/5)*5^(2/5)*Zeta(5)^(7/5)) - Zeta(3)/(3*2^(4/5)*(5*Zeta(5))^(2/5))) * n^(2/5) + (Pi^4/(180*2^(1/5)*(5*Zeta(5))^(3/5))) * n^(3/5) + ((5*(5*Zeta(5))^(1/5))/(2^(8/5))) * n^(4/5)) * Zeta(5)^(67/720) / (2^(113/360) * 5^(293/720) * sqrt(Pi) * n^(427/720)). - Vaclav Kotesovec, Dec 08 2016

A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k-1)/2).

Original entry on oeis.org

1, 1, 10, 40, 155, 560, 2051, 7080, 24064, 79370, 257067, 815593, 2545201, 7812699, 23639459, 70551216, 207932549, 605611061, 1744513262, 4973116444, 14038641287, 39263308551, 108849552289, 299248060986, 816159923366, 2209102273109, 5936069692320, 15840122529455, 41987363787469, 110584436073149

Offset: 0

Ilya Gutkovskiy, Dec 08 2016

nonn

Euler transform of the octagonal pyramidal numbers (A002414).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Index to sequences related to pyramidal numbers

Cf. A000335, A002414, A279215, A279216, A279217, A279218.

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

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k-1)/2).

a(n) ~ exp(-Zeta'(-1)/2 - Zeta(3)/(8*Pi^2) - Pi^16/(671846400000*Zeta(5)^3) - Pi^8*Zeta(3)/(5184000*Zeta(5)^2) - Zeta(3)^2/(240*Zeta(5)) + Zeta'(-3) + (Pi^12/(388800000*2^(3/5)*3^(1/5)*Zeta(5)^(11/5)) + Pi^4*Zeta(3)/(3600*2^(3/5) * 3^(1/5)*Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8/(432000*2^(1/5)*3^(2/5)*Zeta(5)^(7/5)) - Zeta(3)/(2^(11/5)*(3*Zeta(5))^(2/5))) * n^(2/5) + (Pi^4/(180*2^(4/5)*(3*Zeta(5))^(3/5))) * n^(3/5) + ((5*(3*Zeta(5))^(1/5))/(2^(7/5))) * n^(4/5)) * (3*Zeta(5))^(9/100) / (2^(23/100) * sqrt(5*Pi) * n^(59/100)). - Vaclav Kotesovec, Dec 08 2016

A305653 Expansion of Product_{k>=1} 1/(1 - x^k)^((k+1)*binomial(k+2,3)/2).

Original entry on oeis.org

1, 1, 7, 27, 98, 323, 1085, 3471, 10998, 33874, 102737, 305849, 897899, 2597822, 7423408, 20957775, 58524868, 161741013, 442705279, 1200718351, 3228796864, 8611973548, 22793714865, 59887897679, 156252738062, 404964879419, 1042884107691, 2669317020743, 6792321636929

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

Euler transform of A002415, shifted left one place.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A000391, A002415, A023871, A279215, A290792.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d^2*
      (d+2)*(d+1)^2/12, d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 07 2018

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((k + 1) Binomial[k + 2, 3]/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1)^2 (d + 2)/12, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A002415(k+1).
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^5)).
a(n) ~ exp(Zeta'(-1)/6 - Zeta(3) / (4*Pi^2) + 149*Zeta(5) / (32*Pi^4) + 15876 * Zeta(3) * Zeta(5)^2 / Pi^12 - 666792 * Zeta(5)^3 / Pi^14 + 108884466432 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/3 + (-7*(7/2)^(1/6) * Pi / (384*sqrt(3)) - 21 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(3) * Zeta(5) / Pi^7 + 3087 * sqrt(3) * (7/2)^(1/6) * Zeta(5)^2 / (2*Pi^9) - 30339036 * 2^(5/6) * sqrt(3) * 7^(1/6) * Zeta(5)^4 / Pi^19) * n^(1/6) + ((7/2)^(1/3) * Zeta(3) / (2*Pi^2) - 21 * (7/2)^(1/3) * Zeta(5) / (2*Pi^4) + 254016 * 2^(2/3) * 7^(1/3) * Zeta(5)^3 / Pi^14) * n^(1/3) + (sqrt(7/6) * Pi / 12 - 756 * sqrt(42) * Zeta(5)^2 / Pi^9) * sqrt(n) + (9 * 2^(1/3) * 7^(2/3) * Zeta(5) / Pi^4) * n^(2/3) + (2 * (2/7)^(1/6) * sqrt(3) * Pi) / 5 * n^(5/6)) * Pi^(1/90) / (2^(247/270) * 3^(34/45) * 7^(23/270) * n^(79/135)). - Vaclav Kotesovec, Jun 08 2018

A287090 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)^2/4).

Original entry on oeis.org

1, 1, 10, 46, 191, 740, 2912, 10941, 40345, 144703, 509693, 1761738, 5993434, 20076668, 66329914, 216307961, 696990583, 2220665661, 7000973556, 21853019072, 67575353580, 207111103623, 629440843762, 1897670845715, 5677604053474, 16863081962184, 49736388996376, 145714874857754

Offset: 0

Ilya Gutkovskiy, May 19 2017

nonn

Euler transform of A000537.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A000294, A000537, A023872, A279215.

nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k^2 (k + 1)^2/4), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)^2/4).

a(n) ~ exp(-Zeta(3) / (16*Pi^2) + 741*Zeta(5) / (1600*Pi^4) - 250047*Zeta(5)^3 / (5*Pi^14) + 10207918728 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/2 + (-7*(7/2)^(1/6) * Pi / (3200 * 3^(2/3)) + 9261 * 3^(1/3) * (7/2)^(1/6) * Zeta(5)^2 / (40*Pi^9) - 22754277 * 3^(1/3) * (7/2)^(1/6) * Zeta(5)^4 / (2*Pi^19)) * n^(1/6) + (-21 * 3^(2/3) * (7/2)^(1/3) * Zeta(5) / (20*Pi^4) + 31752 * 6^(2/3) * 7^(1/3) * Zeta(5)^3/Pi^14) * n^(1/3) + (sqrt(7/2)*Pi/60 - 567*sqrt(14)*Zeta(5)^2 / Pi^9) * sqrt(n) + 9 * 3^(1/3) * (7/2)^(2/3) * Zeta(5) / Pi^4 * n^(2/3) + 2 * (2/7)^(1/6) * 3^(2/3) * Pi/5 * n^(5/6)) / (2^(1321/1440) * 3^(479/720) * 7^(119/1440) * n^(839/1440) * Pi^(1/240)). - Vaclav Kotesovec, Nov 09 2017

A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(4k^2-1)/3).

Original entry on oeis.org

1, 1, 11, 46, 185, 700, 2676, 9646, 34166, 117500, 396506, 1310527, 4258313, 13607309, 42846151, 133039791, 407833188, 1235202869, 3699140386, 10960888382, 32154531807, 93437164720, 269087234273, 768340525743, 2176098269286, 6115444177489, 17058887661133

Offset: 0

Ilya Gutkovskiy, Apr 08 2018

nonn

Euler transform of A000447.

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms

Cf. A000335, A000447, A023871, A279215.

nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^(k (4 k^2 - 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (4 d^2 - 1)/3, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A000447(k).

a(n) ~ exp(5 * Zeta(5)^(1/5) * n^(4/5)/2 - Zeta(3) * n^(2/5) / (12 * Zeta(5)^(2/5)) + 4*Zeta'(-3)/3 - 1/36 - Zeta(3)^2 / (720*Zeta(5))) * A^(1/3) * Zeta(5)^(83/900) / (2^(7/180) * sqrt(5*Pi) * n^(533/900)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018

A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + x^k)/(k(1 - x^k)^n)).

Original entry on oeis.org

1, 1, 4, 14, 65, 323, 1890, 12002, 83901, 630818, 5081318, 43546333, 395422430, 3788368227, 38151667046, 402516707510, 4436230390977, 50948789415297, 608433141666219, 7540823673023319, 96826154085714992, 1285991546051286085, 17640769457638701839, 249602608552024560609

Offset: 0

Ilya Gutkovskiy, Jun 07 2018

nonn

N. J. A. Sloane, Transforms

Cf. A000990, A023871, A253289, A279215, A293554, A305206, A305653, A305655.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + x^k)/(k (1 - x^k)^n), {k, 1, n}]], {x, 0, n}], {n, 0, 23}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^(2 Binomial[n + k - 2, n - 1] - Binomial[n + k - 3, n - 2]), {k, 1, n}], {x, 0, n}], {n, 0, 23}]

a(n) = [x^n] Product_{k>=1} 1/(1 - x^k)^(2*binomial(n+k-2,n-1)-binomial(n+k-3,n-2)).

A318121 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^4)).

Original entry on oeis.org

1, 1, 4, 15, 65, 269, 1205, 5325, 24064, 108849, 496790, 2275492, 10470720, 48325984, 223721404, 1038182441, 4828274432, 22497132116, 105001996350, 490816448220, 2297356108318, 10766317435860, 50511178395306, 237217429972191, 1115084064063866, 5246116796164594

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

For n > 2, a(n) is the n-th term of the Euler transform of n-gonal pyramidal numbers.

N. J. A. Sloane, Transforms
Index to sequences related to pyramidal numbers

Cf. A000335, A279215, A279216, A279217, A279218, A279219, A318118.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^4), {k, 1, n}]], {x, 0, n}], {n, 0, 25}]

a(n) ~ c * d^n / sqrt(n), where d = 4.80064986801984997726284... and c = 0.244706939300168165858... - Vaclav Kotesovec, Aug 19 2018

A319757 Expansion of Product_{k>=1} (1 - x^k)^(k(k+1)(2*k+1)/6).

Original entry on oeis.org

1, -1, -5, -9, -6, 35, 125, 275, 291, -241, -2111, -5989, -10990, -11660, 6454, 68298, 201859, 400794, 546122, 269907, -1175825, -4890783, -11746437, -20668698, -25146121, -7959643, 63707489, 236244458, 546634684, 956731805, 1220119643, 676723572, -1964409479, -8645307595

Offset: 0

Ilya Gutkovskiy, Sep 27 2018

sign

Cf. A000330, A001157, A001158, A001159, A279215, A281156, A283263, A292386, A292387.

a:=series(mul((1-x^k)^(k*(k+1)*(2*k+1)/6),k=1..100),x=0,34): seq(coeff(a,x,n),n=0..33); # Paolo P. Lava, Apr 02 2019

nmax = 33; CoefficientList[Series[Product[(1 - x^k)^(k (2 k + 1) (k + 1)/6), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[-Sum[x^k (1 + x^k)/(k (1 - x^k)^4), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[d^2 (d + 1) (2 d + 1)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 - x^k)^A000330(k).
G.f.: exp(-Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^4)).
G.f.: exp(-Sum_{k>=1} (2*sigma_4(k) + 3*sigma_3(k) + sigma_2(k))*x^k/(6*k)).

cp's OEIS Frontend

A279216 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)/2).

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A279217 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(4*k-1)/6).

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A279218 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(5*k-2)/6).

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A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(k+1)*(2*k-1)/2).

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A305653 Expansion of Product_{k>=1} 1/(1 - x^k)^((k+1)*binomial(k+2,3)/2).

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A287090 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)^2/4).

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A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(4*k^2-1)/3).

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A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^k)^n)).

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A279217 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(4*k-1)/6).

A279218 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(5*k-2)/6).

A279219 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(k+1)(2*k-1)/2).

A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(4k^2-1)/3).

A305654 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + x^k)/(k(1 - x^k)^n)).

A318121 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^4)).

A319757 Expansion of Product_{k>=1} (1 - x^k)^(k(k+1)(2*k+1)/6).