A280473 -id:A280473 - cp's OEIS frontend

A107742 G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

1, 1, 2, 4, 6, 10, 17, 25, 38, 59, 86, 125, 184, 260, 369, 524, 726, 1005, 1391, 1894, 2576, 3493, 4687, 6272, 8373, 11090, 14647, 19294, 25265, 32991, 42974, 55705, 72025, 92895, 119349, 152965, 195592, 249280, 316991, 402215, 508932, 642598, 809739, 1017850, 1276959, 1599015, 1997943, 2491874, 3102477, 3855165, 4782408, 5922954

Offset: 0

Vladeta Jovovic, Jun 11 2005

From Gus Wiseman, Sep 13 2022: (Start)
Also the number of multiset partitions of integer partitions of n into intervals, where an interval is a set of positive integers with all differences of adjacent elements equal to 1. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1},{1}}  {{1,2}}        {{1},{3}}
                    {{1},{2}}      {{2},{2}}
                    {{1},{1},{1}}  {{1},{1,2}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1},{1}}
Intervals are counted by A001227, ranked by A073485.
The initial version is A007294.
The strict version is A327731.
The version for gapless multisets instead of intervals is A356941.
The case of strict partitions is A356957.
Also the number of multiset partitions of integer partitions of n into distinct constant blocks. For example, the a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}    {{3}}        {{4}}
         {{1,1}}  {{1,1,1}}    {{2,2}}
                  {{1},{2}}    {{1},{3}}
                  {{1},{1,1}}  {{1,1,1,1}}
                               {{2},{1,1}}
                               {{1},{1,1,1}}
Constant multisets are counted by A000005, ranked by A000961.
The non-strict version is A006171.
The unlabeled version is A089259.
The non-constant block version is A261049.
The version for twice-partitions is A279786, factorizations A296131.
Also the number of multiset partitions of integer partitions of n into constant blocks of odd length. For example, a(1) = 1 through a(4) = 6 multiset partitions are:
  {{1}}  {{2}}      {{3}}          {{4}}
         {{1},{1}}  {{1,1,1}}      {{1},{3}}
                    {{1},{2}}      {{2},{2}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1},{1},{2}}
                                   {{1},{1},{1},{1}}
The strict version is A327731 (also).
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A006171, A109386, A219554, A280473, A280486, A288007.
Product_{k>=1} (1 + x^k)^sigma_m(k): this sequence (m=0), A192065 (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).
A000041 counts integer partitions, strict A000009.
A000110 counts set partitions.
A072233 counts partitions by sum and length.
Cf. A001055, A001970, A011782, A055887, A061260, A270995, A279784, A294617, A304969.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j)), {i, 1, nmax}, {j, 1, nmax/i}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 28 2018 *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@chQ/@#&]],{n,0,5}] (* Gus Wiseman, Sep 13 2022 *)

a(n)=polcoeff(prod(k=1,n,prod(j=1,n\k,1+x^(j*k)+x*O(x^n))),n) /* Paul D. Hanna */

N=66;  x='x+O('x^N); gf=1/prod(j=0,N, eta(x^(2*j+1))); gf=prod(j=1,N,(1+x^j)^numdiv(j)); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^(2*m)+x*O(x^n))/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

Euler transform of A001227.
Weigh transform of A000005.
G.f. satisfies: log(A(x)) = Sum_{n>=1} A109386(n)/n*x^n, where A109386(n) = Sum_{d|n} d*Sum_{m|d} (m mod 2). - Paul D. Hanna, Jun 26 2005
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^(2n)) /n ). - Paul D. Hanna, Mar 28 2009
G.f.: Product_{n>=1} Q(x^n) where Q(x) is the g.f. of A000009. - Joerg Arndt, Feb 27 2014
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 04 2017
Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/6). - Vaclav Kotesovec, Aug 29 2018

More terms from Paul D. Hanna, Jun 26 2005

A174465 G.f.: exp( Sum_{n>=1} A174466(n)x^n/n ) where A174466(n) = Sum_{d|n} dsigma(n/d)*tau(d).

Original entry on oeis.org

1, 1, 4, 7, 19, 31, 74, 122, 258, 430, 835, 1378, 2557, 4162, 7382, 11932, 20471, 32676, 54634, 86251, 141001, 220371, 353413, 546783, 863043, 1322425, 2057525, 3125092, 4801297, 7230393, 10984924, 16410474, 24679719, 36593278, 54526145, 80272501

Offset: 0

Paul D. Hanna, Apr 04 2010

nonn

Compare to the g.f. of the number of planar partitions of n (A000219):
exp( Sum_{n>=1} sigma_2(n)*x^n/n ) where sigma_2(n) = Sum_{d|n} d*sigma(n/d)*phi(d).
tau(n) = A000005(n) = the number of divisors of n,
and sigma(n) = A000203(n) = sum of divisors of n.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A174466, A000203 (sigma), A000005 (tau), A006171, A007425, A280473, A280487.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(i*j*k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 04 2017 *)
nmax = 50; A007425 = Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[A007425[[k]], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 30 2018 *)

{a(n)=polcoeff(exp(sum(m=1,n,x^m/m*sumdiv(m,d,d*sigma(m/d)*sigma(d,0)))+x*O(x^n)),n)}

G.f.: Product_{i>=1, j>=1, k>=1} 1/(1 - x^(i*j*k)). - Vaclav Kotesovec, Jan 04 2017

G.f.: Product_{k>=1} 1/(1 - x^k)^tau_3(k), where tau_3() = A007425. - Ilya Gutkovskiy, May 22 2018

A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijk*l)).

Original entry on oeis.org

1, 1, 4, 8, 20, 36, 86, 150, 314, 564, 1088, 1902, 3557, 6085, 10902, 18506, 32124, 53584, 91133, 149749, 249315, 405121, 662582, 1063152, 1714580, 2719842, 4327302, 6797316, 10686005, 16622003, 25861855, 39866017, 61422891, 93910783, 143406552, 217537696

Offset: 0

Vaclav Kotesovec, Jan 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000009, A007426, A107742, A280473, A280487, A219561.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j*k*l)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}], {x, 0, nmax}], x]
nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#] * DivisorSigma[0, #] &], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[tau4[[k]], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Sep 08 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^tau_4(k), where tau_4() = A007426. - Ilya Gutkovskiy, May 22 2018

A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))/(1 - x^(ijk)).

Original entry on oeis.org

1, 2, 8, 20, 56, 128, 316, 684, 1532, 3192, 6704, 13436, 26984, 52352, 101316, 191320, 359334, 662292, 1213360, 2189380, 3925432, 6951592, 12231332, 21298452, 36856840, 63211164, 107765896, 182295468, 306625208, 512190992, 851011960, 1405199028, 2308629300, 3771593392

Offset: 0

Ilya Gutkovskiy, May 24 2018

nonn

Convolution of the sequences A174465 and A280473.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Index entries for sequences related to partitions

Cf. A007425, A174465, A280473, A301554.

nmax = 33; CoefficientList[Series[Product[(1 + x^(i j k))/(1 - x^(i j k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A007425(k).

A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))^(ijk).

Original entry on oeis.org

1, 1, 6, 15, 48, 108, 323, 716, 1868, 4217, 10137, 22311, 51477, 110817, 245260, 519918, 1114914, 2318557, 4854952, 9923533, 20335761, 40941170, 82365742, 163413699, 323589060, 633429923, 1236392498, 2390718266, 4606489839, 8805346615, 16768968317, 31713677061, 59747953446

Offset: 0

Ilya Gutkovskiy, Aug 26 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A007425, A034718, A280473, A280541, A318413.

a:=series(mul(mul(mul((1+x^(i*j*k))^(i*j*k),k=1..55),j=1..55),i=1..55),x=0,33): seq(coeff(a,x,n),n=0..32); # Paolo P. Lava, Apr 02 2019

nmax = 32; CoefficientList[Series[Product[Product[Product[(1 + x^(i j k))^(i j k), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Product[(1 + x^k)^(k Sum[DivisorSigma[0, d], {d, Divisors[k]}]), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}]  x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
nmax = 32; A034718 = Table[n*Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[A034718[[k]], j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]]; , {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] (* Vaclav Kotesovec, Aug 31 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^(k*tau_3(k)), where tau_3() = A007425.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d^2 * Sum_{j|d} tau(j) ) * x^k/k), where tau() = A000005.
Conjecture: log(a(n)) ~ 3^(2/3) * Zeta(3)^(1/3) * log(n)^(2/3) * n^(2/3) / 2^(5/3). - Vaclav Kotesovec, Sep 02 2018

A321359 Expansion of Product_{1 <= i <= j <= k} (1 + x^(ijk)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 11, 16, 21, 27, 38, 49, 63, 84, 108, 137, 179, 226, 286, 365, 457, 570, 720, 894, 1106, 1378, 1700, 2087, 2577, 3151, 3847, 4707, 5723, 6941, 8439, 10197, 12300, 14852, 17863, 21433, 25740, 30797, 36794, 43963, 52372, 62288, 74098, 87905, 104149

Offset: 0

Seiichi Manyama, Nov 07 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A034836, A211856, A280473, A321360, A321361.

G.f.: Product_{k>0} (1 + x^k)^A034836(k).

A321192 a(n) = [x^n] Product_{k>=1} (1 + x^k)^tau_n(k), where tau_n(k) = number of ordered n-factorizations of k.

Original entry on oeis.org

1, 1, 2, 6, 20, 55, 239, 700, 3212, 10104, 48622, 161579, 806843, 2799199, 14379647, 52018828, 273472712, 1023655306, 5491615463, 21234676241, 115910309103, 460998296937, 2556361045845, 10440651927427, 58714921974979, 245586789818255, 1399187406060485

Offset: 0

Ilya Gutkovskiy, Oct 29 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A000009, A077592, A107742, A280473, A280486, A321191.

tau[n_, 1] = 1; tau[n_, k_] := tau[n, k] = Plus @@ (tau[#, k-1] & /@ Divisors[n]); nmax = 30; Table[SeriesCoefficient[Product[(1 + x^k)^tau[k, n], {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Oct 29 2018 *)

a(n) = [x^n] Product_{k_1>=1, k_2>=1, ..., k_n>=1} (1 + x^(k_1*k_2*...*k_n)).

A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(ijk)).

Original entry on oeis.org

1, 1, 4, 10, 31, 82, 241, 664, 1898, 5316, 15058, 42374, 119718, 337432, 952373, 2685906, 7578248, 21376331, 60306495, 170120330, 479922212, 1353855927, 3819280961, 10774233218, 30394408336, 85743168417, 241883489742, 682358211402, 1924947591447, 5430317571250, 15319043353639

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A007425.

N. J. A. Sloane, Transforms
Index entries for sequences related to compositions

Cf. A000005, A007425, A011782, A129921, A174465, A280473, A304964.

A:= proc(n, k) option remember; `if`(k=1, 1,
      add(A(d, k-1), d=numtheory[divisors](n)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(A(j, 3)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018

nmax = 30; CoefficientList[Series[1/(1 - Sum[x^(i j k), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}]), {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[1/(1 - Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Sum[DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 30}]

G.f.: 1/(1 - Sum_{k>=1} A007425(k)*x^k).

A318967 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))^(1/(ijk)).

Original entry on oeis.org

1, 1, 3, 15, 69, 477, 4167, 34731, 333225, 4058073, 48535659, 638782119, 9690930477, 146665611765, 2428164153711, 44904494549763, 820664075440593, 16238018609968689, 350155700132388435, 7568774583230565567, 175171222712837235861, 4318996957424273510541, 107317465474650443023383

Offset: 0

Ilya Gutkovskiy, Sep 06 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..444
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A007425, A168243, A280473, A318414, A318696, A318768, A318966.

a:=series(mul(mul(mul((1+x^(i*j*k))^(1/(i*j*k)),k=1..55),j=1..55),i=1..55),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Apr 02 2019

nmax = 22; CoefficientList[Series[Product[Product[Product[(1 + x^(i j k))^(1/(i j k)), {i, 1, nmax}], {j, 1, nmax}], {k, 1, nmax} ], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(Sum[DivisorSigma[0, d], {d, Divisors[k]}]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = (n - 1)! Sum[Sum[(-1)^(k/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]

E.g.f.: Product_{k>=1} (1 + x^k)^(tau_3(k)/k), where tau_3 = A007425.

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

A319359 Expansion of Product_{i>=1, j>=1, k>=1} (1 - x^(ijk)).

Original entry on oeis.org

1, -1, -3, 0, 0, 9, 5, 4, -1, -27, -2, -33, -17, 8, 43, 92, 36, 100, -8, -11, -136, -120, -296, -363, -13, -203, 286, 306, 1010, 667, 724, 790, 151, -258, -1207, -964, -3325, -2059, -2924, -1992, -2116, 1277, 3625, 4437, 7724, 7734, 11524, 5801, 9685, -855, -2799, -13409, -16423

Offset: 0

Ilya Gutkovskiy, Sep 17 2018

sign

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000005, A000203, A007425, A174465, A174466, A280473, A288098.

a:=series(mul(mul(mul(1-x^(i*j*k),k=1..55),j=1..55),i=1..55),x=0,53): seq(coeff(a,x,n),n=0..52); # Paolo P. Lava, Apr 02 2019

nmax = 52; CoefficientList[Series[Product[(1 - x^(i j k)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}], {x, 0, nmax}], x]
nmax = 52; CoefficientList[Series[Product[(1 - x^k)^Sum[DivisorSigma[0, d], {d, Divisors[k]}], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[d DivisorSigma[1, k/d] DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 52}]

G.f.: Product_{k>=1} (1 - x^k)^A007425(k).

G.f.: exp(-Sum_{k>=1} A174466(k)*x^k/k).

cp's OEIS Frontend

A107742 G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).

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A174465 G.f.: exp( Sum_{n>=1} A174466(n)*x^n/n ) where A174466(n) = Sum_{d|n} d*sigma(n/d)*tau(d).

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A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(i*j*k*l)).

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A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))/(1 - x^(i*j*k)).

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A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^(i*j*k).

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A321359 Expansion of Product_{1 <= i <= j <= k} (1 + x^(i*j*k)).

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A321192 a(n) = [x^n] Product_{k>=1} (1 + x^k)^tau_n(k), where tau_n(k) = number of ordered n-factorizations of k.

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A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(i*j*k)).

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A174465 G.f.: exp( Sum_{n>=1} A174466(n)x^n/n ) where A174466(n) = Sum_{d|n} dsigma(n/d)*tau(d).

A280486 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijk*l)).

A305050 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))/(1 - x^(ijk)).

A318414 Expansion of Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))^(ijk).

A321359 Expansion of Product_{1 <= i <= j <= k} (1 + x^(ijk)).

A304963 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1} x^(ijk)).

A318967 Expansion of e.g.f. Product_{i>=1, j>=1, k>=1} (1 + x^(ijk))^(1/(ijk)).

A319359 Expansion of Product_{i>=1, j>=1, k>=1} (1 - x^(ijk)).