A023872 -id:A023872 - cp's OEIS frontend

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

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.

A343283 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k^3).

Original entry on oeis.org

1, 8, 27, 100, 125, 432, 343, 1144, 1107, 2000, 1331, 6156, 2197, 5488, 6750, 12906, 4913, 20520, 6859, 28500, 18522, 21296, 12167, 80136, 23500, 35152, 43020, 78204, 24389, 135000, 29791, 141848, 71874, 78608, 85750, 320760, 50653, 109744, 118638, 371000, 68921, 370440, 79507

Offset: 1

Ilya Gutkovskiy, Apr 10 2021

nonn

Cf. A000578, A001055, A023872, A050367, A329125, A343284, A343285, A343286, A343287, A343288, A343289.

A300975 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^3))^n.

Original entry on oeis.org

1, 1, 3, 10, 35, 126, 462, 1716, 6443, 24391, 92928, 355862, 1368458, 5280744, 20438148, 79302960, 308385355, 1201536286, 4689450021, 18330233110, 71747534460, 281177705490, 1103163479190, 4332522733560, 17031238725410, 67007449610751, 263841039245280, 1039628691988795

Offset: 0

Ilya Gutkovskiy, Mar 17 2018

nonn

Number of partitions of n into cubes of n kinds.

Cf. A000578, A003108, A008485, A023872, A298434, A300974.

a:= proc(m) option remember; local b; b:= proc(n, i)
      option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(m+j-1, j)*b(n-i^3*j, i-1), j=0..n/i^3)))
      end: b(n, iroot(n, 3))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2018

Table[SeriesCoefficient[Product[1/(1 - x^k^3)^n, {k, 1, n}], {x, 0, n}], {n, 0, 27}]

a(n) ~ c * d^n / sqrt(n), where d = 4.0147940395164236614815683662796167488... and c = 0.2726202310726337579308600184572222... - Vaclav Kotesovec, Mar 23 2018

A304460 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^3).

Original entry on oeis.org

1, 1, 44, 4410, 905840, 318906400, 172185088824, 132357574570221, 137406570363495360, 185242628827767255255, 314645673306845990409300, 657405676947400829561901359, 1656968286301847988118098735168, 4957425610652588047512198547937050

Offset: 0

Vaclav Kotesovec, May 13 2018

nonn

In general, for m>=3, coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^m) is asymptotic to n^(m*n)/n!.

Cf. A008485, A023872, A304446, A304459.

nmax = 20; Table[SeriesCoefficient[Product[1/(1-x^k)^(n^3), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[1/QPochhammer[x]^(n^3), {x, 0, n}], {n, 0, nmax}]

a(n) ~ exp(n) * n^(2*n - 1/2) / sqrt(2*Pi).

A328408 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4.

Original entry on oeis.org

1, 9, 27, 73, 125, 243, 343, 585, 729, 1125, 1331, 1971, 2197, 3087, 3375, 4681, 4913, 6561, 6859, 9125, 9261, 11979, 12167, 15795, 15625, 19773, 19683, 25039, 24389, 30375, 29791, 37449, 35937, 44217, 42875, 53217, 50653, 61731, 59319, 73125, 68921, 83349, 79507, 97163, 91125

Offset: 1

Ilya Gutkovskiy, Oct 14 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000578, A001511, A016755, A023872, A059376, A129527, A209229, A328407.

[n eq 1 select 1 else IsOdd(n) select n^3 else Self(n div 2)+n^3 :n in [1..45]]; // Marius A. Burtea, Oct 15 2019

nmax = 45; CoefficientList[Series[Sum[x^(2^k) (1 + 4 x^(2^k) + x^(2^(k + 1)))/(1 - x^(2^k))^4, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := If[EvenQ[n], a[n/2] + n^3, n^3]; Table[a[n], {n, 1, 45}]
Table[DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] #^3 &], {n, 1, 45}]
f[p_, e_] :=p^(3*e); f[2, e_] := (8^(e+1)-1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 23 2023 *)

G.f.: Sum_{k>=0} x^(2^k) * (1 + 4*x^(2^k) + x^(2^(k+1))) / (1 - x^(2^k))^4.
G.f.: (1/7) * Sum_{k>=1} J_3(2*k) * x^k / (1 - x^k), where J_3() is the Jordan function (A059376).
Dirichlet g.f.: zeta(s-3) / (1 - 2^(-s)).
a(2*n) = a(n) + 8*n^3, a(2*n+1) = (2*n + 1)^3.
a(n) = Sum_{d|n} A209229(n/d) * d^3.
Product_{n>=1} (1 + x^n)^a(n) = g.f. for A023872.
Sum_{k=1..n} a(k) ~ 4*n^4/15. - Vaclav Kotesovec, Oct 15 2019
Multiplicative with a(2^e) = (8^(e+1)-1)/7, and a(p^e) = p^(3*e) for an odd prime p. - Amiram Eldar, Oct 23 2023

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

A381709 Euler transform of n^3 * A065960(n).

Original entry on oeis.org

1, 1, 137, 2351, 29075, 408429, 5957562, 76590384, 955079422, 11831378688, 142650905559, 1668991927795, 19144774189917, 215790313316371, 2388025355854986, 25973791505651972, 278176027053878678, 2936495245822593766, 30573379794788083289, 314185573464039742503

Offset: 0

Seiichi Manyama, Mar 04 2025

nonn

Cf. A156303, A156733, A381170.

Cf. A023872, A065960, A381679.

my(N=20, x='x+O('x^N)); Vec(exp(sum(k=1, N, sigma(k^2, 4)*x^k/k)))

G.f.: 1/Product_{k>=1} (1 - x^k)^(k^3 * A065960(k)).
G.f.: exp( Sum_{k>=1} sigma_4(k^2) * x^k/k ).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} sigma_4(k^2) * a(n-k).

A279761 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k^2+1)/3).

Original entry on oeis.org

1, 1, 7, 26, 91, 290, 946, 2922, 8937, 26521, 77485, 222005, 626988, 1743739, 4787625, 12979799, 34792728, 92257673, 242197348, 629805075, 1623197726, 4148192991, 10516418844, 26458470616, 66086152465, 163925621199, 403931474096, 989040788801, 2407020523315, 5823830868091, 14011949899801, 33530477120905, 79820957945103

Offset: 0

Ilya Gutkovskiy, Dec 18 2016

nonn

Euler transform of the octahedral numbers (A005900).

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, Octahedral Number
OEIS Wiki, Platonic numbers

Cf. A000335, A005900, A023872.

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

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

a(n) ~ exp(Zeta'(-1)/3 - Zeta(3)^2 / (360*Zeta(5)) + 2*Zeta'(-3)/3 + (Zeta(3)/(6*2^(3/5) * Zeta(5)^(2/5))) * n^(2/5) + (5*(Zeta(5)/2)^(1/5)/2) * n^(4/5)) * Zeta(5)^(47/450) / (2^(37/450) * sqrt(5*Pi) * n^(136/225)). - Vaclav Kotesovec, Nov 09 2017

A279762 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k^2-5*k+2)/2).

Original entry on oeis.org

1, 1, 13, 61, 263, 1094, 4578, 18076, 69815, 262242, 965342, 3480006, 12322360, 42896002, 147062818, 497000146, 1657470977, 5459160063, 17772284155, 57225458626, 182362100816, 575463112191, 1799106136923, 5575063264825, 17130798464652, 52216240087807, 157937816918539, 474197830869573, 1413695306175884, 4185962563381518

Offset: 0

Ilya Gutkovskiy, Dec 18 2016

nonn

Euler transform of the icosahedral numbers (A006564).

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
OEIS Wiki, Platonic numbers

Cf. A000335, A006564, A023872.

nmax=29; CoefficientList[Series[Product[1/(1 - x^k)^(k (5 k^2 - 5 k + 2)/2), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ exp(Zeta'(-1) + 5*Zeta(3) / (8*Pi^2) - Pi^16 / (16796160000*Zeta(5)^3) + Pi^8 * Zeta(3) / (648000*Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + 5*Zeta'(-3)/2 + (-Pi^12/(19440000 * 2^(2/5) * 15^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (21600 * 2^(4/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(4/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + (-Pi^4 / (36 * 2^(1/5) * (15*Zeta(5))^(3/5))) * n^(3/5) + ((5*(15*Zeta(5))^(1/5)) / 2^(8/5)) * n^(4/5)) * (3*Zeta(5))^(9/80) / (2^(11/40) * 5^(31/80) * sqrt(Pi) * n^(49/80)). - Vaclav Kotesovec, Nov 09 2017

A279763 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-1)(3k-2)/2).

Original entry on oeis.org

1, 1, 21, 105, 535, 2670, 12996, 59546, 266875, 1161894, 4939778, 20528320, 83636061, 334496221, 1315381029, 5091782355, 19424086781, 73092029218, 271537720562, 996656173345, 3616680935702, 12983391870459, 46133749660407, 162337625047433, 565962994479384, 1955721907216420, 6701061533668542, 22774651422340672

Offset: 0

Ilya Gutkovskiy, Dec 18 2016

nonn

Euler transform of the dodecahedral numbers (A006566).

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
OEIS Wiki, Platonic numbers

Cf. A000335, A006566, A023872.

nmax=27; CoefficientList[Series[Product[1/(1 - x^k)^(k (3 k - 1) (3 k - 2)/2), {k, 1, nmax}], {x, 0, nmax}], x]

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

a(n) ~ exp(Zeta'(-1) + 9*Zeta(3) / (8*Pi^2) - Pi^16 / (9331200000*Zeta(5)^3) + Pi^8 * Zeta(3) / (648000*Zeta(5)^2) - Zeta(3)^2 / (270*Zeta(5)) + 9*Zeta'(-3)/2 + (-Pi^12/(10800000 * 2^(2/5) * 3^(3/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * 3^(3/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (36000 * 2^(4/5) * 3^(1/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(4/5) * 3^(6/5) * Zeta(5)^(2/5))) * n^(2/5) + (-Pi^4 / (60 * 2^(1/5) * 3^(4/5) * Zeta(5)^(3/5))) * n^(3/5) + ((5*3^(3/5) * Zeta(5)^(1/5)) / 2^(8/5)) * n^(4/5)) * 3^(131/400) * Zeta(5)^(131/1200) / (2^(169/600) * sqrt(5*Pi) * n^(731/1200)). - Vaclav Kotesovec, Nov 09 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A343283 Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^(k^3).

Views

Author

Keywords

Crossrefs

A300975 a(n) = [x^n] Product_{k>=1} 1/(1 - x^(k^3))^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A304460 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(n^3).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A328408 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + 4*x + x^2) / (1 - x)^4.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A381709 Euler transform of n^3 * A065960(n).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A279761 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(2*k^2+1)/3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A279762 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(5*k^2-5*k+2)/2).

Views

Author

Keywords

Comments

Links

Crossrefs

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

A279761 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k^2+1)/3).

A279762 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k^2-5*k+2)/2).

A279763 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-1)(3k-2)/2).