A000335 -id:A000335 - cp's OEIS frontend

A258350 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k+1)(k+2)).

1, 6, 45, 260, 1410, 7026, 33212, 149190, 643959, 2681020, 10820736, 42468828, 162566956, 608302638, 2229485529, 8016901068, 28324233846, 98447346282, 336996263702, 1137220855428, 3786525025002, 12449461237388, 40446207528429, 129926295916884, 412912082761651

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A023872, A000335, A258342, A258347, A258348, A258349, A258351, A258352.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)*(k+2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(79/600) / (2^(21/200) * sqrt(5*Pi) * n^(379/600)) * exp(2*Zeta'(-1) - 3*Zeta(3)/(4*Pi^2) - Pi^16 / (518400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3) + (Pi^12 / (1800000 * 2^(3/5) * 3^(1/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (150 * 2^(3/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(1/5) * 3^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(1/5) * (3*Zeta(5))^(2/5))) * n^(2/5) + Pi^4 / (30 * 2^(4/5) * (3*Zeta(5))^(3/5)) * n^(3/5) + 5 * (3*Zeta(5))^(1/5) / 2^(7/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

A258351 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 141, 354, 996, 2720, 7194, 18306, 46154, 115506, 288195, 713210, 1749732, 4253148, 10259302, 24573390, 58491312, 138371354, 325415727, 760899396, 1769420183, 4093054602, 9420739965, 21578842582, 49199229066, 111672215658, 252381169048

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000335, A023872, A258345, A258347, A258348, A258349, A258350, A258352.

nmax=40; CoefficientList[Series[Product[1/(1-x^k)^(k*(k-1)*(k-2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(79/600) / (2^(21/200) * sqrt(5*Pi) * n^(379/600)) * exp(2*Zeta'(-1) + 3*Zeta(3)/(4*Pi^2) - Pi^16 / (518400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3) + (-Pi^12 / (1800000 * 2^(3/5) * 3^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (150 * 2^(3/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(1/5) * 3^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(1/5) * (3*Zeta(5))^(2/5))) * n^(2/5) - Pi^4 / (30 * 2^(4/5) * (3*Zeta(5))^(3/5)) * n^(3/5) + 5 * (3*Zeta(5))^(1/5) / 2^(7/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

A278768 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-1)/2).

Original entry on oeis.org

1, 1, 6, 18, 55, 150, 424, 1113, 2923, 7401, 18510, 45271, 109297, 259447, 608428, 1407958, 3222132, 7292198, 16340830, 36265672, 79775931, 173999194, 376497975, 808471181, 1723592762, 3649271887, 7675809680, 16043777217, 33332888108, 68853608216, 141438908854, 288994878713, 587458691042

Offset: 0

Ilya Gutkovskiy, Nov 28 2016

nonn

Euler transform of the pentagonal numbers (A000326).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
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 Number
Index to sequences related to polygonal numbers

Cf. A000294, A000326, A000335, A023871.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(3*d-1)/2, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (3 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(3*k-1)/2).

a(n) ~ exp(-Zeta'(-1)/2 - 3*Zeta(3)/(8*Pi^2) - 25*Zeta(3)^3/(6*Pi^8) - 5^(5/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - sqrt(5/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(7/4)*Pi/(3*5^(1/4)) * n^(3/4)) / (2^(155/96) * 5^(11/96) * Pi^(1/24) * n^(59/96)). - Vaclav Kotesovec, Dec 02 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

A007327 Difference between two partition g.f.s.

Original entry on oeis.org

0, 0, 0, 0, 0, 5, 20, 69, 200, 521, 1294, 3126, 7364, 17309, 40577, 95460, 224971, 531368, 1252664, 2943095, 6870029, 15911618, 36507381, 82930347, 186414619, 414654766, 912766795, 1989007381, 4292038414, 9175624264, 19442250125, 40851448761, 85157787033, 176200110937

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

George E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 190.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100; alternative link.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Cf. A000334, A000335.

Cf. A007326, A007328, A007329, A007330, A008780, A042984.

a(n) = A000335(n) - A000334(n). - Sean A. Irvine, Dec 18 2017

a(11)-a(23) from Sean A. Irvine, Dec 18 2017

More terms from Amiram Eldar, May 11 2024

A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-2)).

Original entry on oeis.org

1, 1, 9, 30, 106, 339, 1106, 3355, 10102, 29358, 83908, 234394, 644286, 1739933, 4631675, 12153197, 31485413, 80576160, 203902261, 510490213, 1265353568, 3106771717, 7559844833, 18239351931, 43650061720, 103657177941, 244346681972, 571930478187, 1329655624297, 3071230379625, 7049750442386, 16085170634548, 36489192684910

Offset: 0

Ilya Gutkovskiy, Nov 30 2016

Euler transform of the octagonal numbers (A000567).

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
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, Octagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000567, A000335, A023871, A278768, A294667, A294691, A294692.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(3*d-2), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]

from sympy import divisors
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n==0 else sum(sum(d**2*(3*d - 2) for d in divisors(j))*a(n - j) for j in range(1, n + 1))//n
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 06 2017, after Maple code

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)).

a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) - 2*Zeta(3) * sqrt(5*n) / Pi^2 - 10*Zeta(3)^2 * (5*n)^(1/4) / Pi^5 - 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) - 1/6) * A^2 / (2^(3/2) * 5^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017

A264924 G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)(n+2)(n+3)/3!).

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 10, 20, 36, 60, 104, 180, 336, 620, 1174, 2160, 3961, 7100, 12690, 22424, 39651, 69820, 122970, 215904, 378470, 660872, 1150740, 1996200, 3452685, 5952916, 10237576, 17559460, 30049285, 51301020, 87390872, 148534232, 251916041, 426329040, 720003646, 1213481344, 2041155052, 3426721080

Offset: 0

Paul D. Hanna, Nov 28 2015

nonn

Number of partitions of n objects of 4 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

			G.f.: A(x) = 1 + x^4 + 4*x^5 + 10*x^6 + 20*x^7 + 36*x^8 + 60*x^9 + 104*x^10 + 180*x^11 +...
where
1/A(x) = (1-x^4) * (1-x^5)^4 * (1-x^6)^10 * (1-x^7)^20 * (1-x^8)^35 * (1-x^9)^56 * (1-x^10)^84 * (1-x^11)^120 * (1-x^12)^165 *...
Also,
log(A(x)) = (x/(1-x))^4 + (x^2/(1-x^2))^4/2 + (x^3/(1-x^3))^4/3 + (x^4/(1-x^4))^4/4 + (x^5/(1-x^5))^4/5 + (x^6/(1-x^6))^4/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A052847, A264923, A264925, A264926.

Cf. A000335, A255050, A258352.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-3)*(k-2)*(k-1)/6), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+4) +x*O(x^n) )^((k+1)*(k+2)*(k+3)/3!) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^4 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{L(n) = sumdiv(n,d, d*(d-1)*(d-2)*(d-3)/3! )}
{a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^4 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)*(d-3)/3!.
a(n) ~ Zeta(5)^(109/3600) / (2^(791/1800) * n^(1909/3600) * sqrt(5*Pi)) * exp(11*Zeta'(-1)/6 + log(2*Pi)/2 + Zeta(3)/(4*Pi^2) - Pi^16/(194400000 * Zeta(5)^3) + 11*Pi^8 * Zeta(3)/(108000 * Zeta(5)^2) - Pi^6/(1800*Zeta(5)) - 121*Zeta(3)^2/(360*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12/(1350000 * 2^(2/5) * Zeta(5)^(11/5)) + 11*Pi^4 * Zeta(3)/(900 * 2^(2/5) * Zeta(5)^(6/5)) - Pi^2/(3*2^(7/5) * Zeta(5)^(1/5))) * n^(1/5) + (-Pi^8/(9000 * 2^(4/5) * Zeta(5)^(7/5)) + 11*Zeta(3)/(3*2^(9/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4/(90 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5*Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)). - Vaclav Kotesovec, Dec 09 2015

A344203 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^binomial(k+2,3).

Original entry on oeis.org

1, 4, 10, 30, 35, 96, 84, 220, 220, 360, 286, 888, 455, 896, 1030, 1741, 969, 2580, 1330, 3470, 2611, 3168, 2300, 7936, 3555, 5096, 5524, 8820, 4495, 13240, 5456, 14144, 9405, 11016, 10710, 27072, 9139, 15200, 15210, 31940, 12341, 33992, 14190, 31856, 30715

Offset: 1

Ilya Gutkovskiy, May 11 2021

nonn

Cf. A000292, A000335, A001055, A050367, A329124, A344204, A344205, A344206, A344207.

cp's OEIS Frontend

A258350 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k+1)*(k+2)).

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A258351 Expansion of Product_{k>=1} 1/(1-x^k)^(k*(k-1)*(k-2)).

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A278768 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*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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A007327 Difference between two partition g.f.s.

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A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)).

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A258350 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k+1)(k+2)).

A258351 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)).

A278768 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-1)/2).

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).

A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-2)).

A264924 G.f.: 1 / Product_{n>=0} (1 - x^(n+4))^((n+1)(n+2)(n+3)/3!).