A107742 -id:A107742 - cp's OEIS frontend

A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)x^n = log( Sum_{n>=0} A107742(n)x^n ).

1, 3, 7, 7, 11, 21, 15, 15, 34, 33, 23, 49, 27, 45, 77, 31, 35, 102, 39, 77, 105, 69, 47, 105, 86, 81, 142, 105, 59, 231, 63, 63, 161, 105, 165, 238, 75, 117, 189, 165, 83, 315, 87, 161, 374, 141, 95, 217, 162, 258, 245, 189, 107, 426, 253, 225, 273, 177, 119, 539, 123, 189, 510, 127, 297

Offset: 1

Paul D. Hanna, Jun 26 2005

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000005, A001227, A060640, A107742.

Sum_{d|n} d^k*A000593(n/d): A288417 (k=0), this sequence (k=1), A288418 (k=2), A288419 (k=3), A288420 (k=4).

a[n_] := DivisorSum[n, #*DivisorSum[#, Mod[#, 2]&]&]; Array[a, 65] (* Jean-François Alcover, Dec 23 2015 *)
f[p_, e_] := ((p + e*(p-1) - 2)*p^(e+1) + 1)/(p-1)^2; f[2, e_] := 2^(e+1) - 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 29 2023 *)

PARI
```
a(n)=sumdiv(n,d,d*sumdiv(d,m,m%2))
```

N=66; x='x+O('x^N); /* that many terms */
c=sum(j=1, N, j*x^j);
t=log( 1/prod(j=0, N, eta(x^(2*j+1))) );
gf=serconvol(t, c);
Vec(gf) /* show terms */
/* Joerg Arndt, May 03 2008 */

a(n) = Sum_{d|n} d * Sum_{m|d} (m mod 2).
G.f.: Sum_{n>=1} a(n)/n*x^n = Sum_{j>=1} Sum_{i>=1} log(1+x^(i*j)).
From Vladeta Jovovic, Jul 05 2005:(Start)
Multiplicative with a(2^e) = 2^(e+1)-1 and a(p^e) = (p^(e+2)*(e+1)-p^(e+1)*(e+2)+1)/(p-1)^2 for p>2.
G.f.: Sum_{n>0} n*A000005(n)*x^n/(1+x^n).
G.f.: Sum_{n>0} n*A001227(n)*x^n/(1-x^n).
a(n) = A060640(n) if n is odd, else a(n) = A060640(n) - 2*A060640(n/2).
a(n) = Sum_{d|n} d*A001227(d).
a(n) = Sum_{d|n} d*A000593(n/d).
A107742(n) = (1/n)*Sum_{k=1..n} a(k)*A107742(n-k). (End)

A192065 Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 1, 3, 7, 14, 28, 58, 106, 201, 372, 669, 1187, 2101, 3624, 6229, 10591, 17796, 29659, 49107, 80492, 131157, 212237, 341084, 544883, 865717, 1367233, 2148552, 3359490, 5227270, 8096544, 12486800, 19174319, 29326306, 44678825, 67811375, 102549673, 154545549

Offset: 0

Joerg Arndt, Jun 24 2011

nonn

Euler transform of A002131. - Vaclav Kotesovec, Mar 26 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
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. A061256 (1/Product_{k>=1} P(x^k)^k where P(x) = Product_{k>=1} (1 - x^k)).

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), this sequence (m=1), A288414 (m=2), A288415 (m=3), A301548 (m=4), A301549 (m=5), A301550 (m=6), A301551 (m=7), A301552 (m=8).

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[(1 + x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)
kmax = 37; Product[QPochhammer[-1, x^k]^k/2^k, {k, 1, kmax}] + O[x]^kmax // CoefficientList[#, x]& (* Jean-François Alcover, Jul 03 2017 *)
nmax = 40; CoefficientList[Series[Exp[Sum[Sum[DivisorSum[k, # / GCD[#, 2] &] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

N=66;  x='x+O('x^N);
Q(x)=prod(k=1,N,1+x^k);
gf=prod(k=1,N, Q(x^k)^k );
Vec(gf) /* Joerg Arndt, Jun 24 2011 */

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0. - Seiichi Manyama, Jun 09 2017

a(n) ~ exp(3*Pi^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2^(5/3) - Pi^(4/3) * n^(1/3) / (3*2^(7/3) * Zeta(3)^(1/3)) - Pi^2 / (864 * Zeta(3))) * Zeta(3)^(1/6) / (2^(19/24) * sqrt(3) * Pi^(1/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018

A301554 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma_0(k)).

Original entry on oeis.org

1, 2, 6, 14, 32, 66, 138, 266, 512, 948, 1730, 3074, 5408, 9306, 15854, 26594, 44150, 72378, 117620, 189074, 301516, 476518, 747514, 1163470, 1798920, 2762040, 4215194, 6393196, 9642596, 14462518, 21581386, 32040562, 47345342, 69635866, 101974722, 148692638

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Convolution of A006171 and A107742.

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

Cf. A000005, A006171, A107742, A320237.

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

with(numtheory): seq(coeff(series(mul(((1+x^k)/(1-x^k))^sigma[0](k),k=1..n),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 29 2018

nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

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

G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))/(1 - x^(i*j)). - Ilya Gutkovskiy, May 23 2018

Conjecture: log(a(n)) ~ Pi * sqrt(n*log(n)/2). - Vaclav Kotesovec, Sep 03 2018

A356932 Number of multiset partitions of integer partitions of n such that all blocks have odd size.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 24, 42, 74, 130, 224, 383, 653, 1100, 1846, 3079, 5104, 8418, 13827, 22592, 36774, 59613, 96271, 154908, 248441, 397110, 632823, 1005445, 1592962, 2516905, 3966474, 6235107, 9777791, 15297678, 23880160, 37196958, 57819018, 89691934, 138862937

Offset: 0

Gus Wiseman, Sep 11 2022

nonn

			The a(1) = 1 through a(5) = 13 multiset partitions:
  {1}  {2}     {3}        {4}           {5}
       {1}{1}  {111}      {112}         {113}
               {1}{2}     {1}{3}        {122}
               {1}{1}{1}  {2}{2}        {1}{4}
                          {1}{111}      {2}{3}
                          {1}{1}{2}     {11111}
                          {1}{1}{1}{1}  {1}{112}
                                        {2}{111}
                                        {1}{1}{3}
                                        {1}{2}{2}
                                        {1}{1}{111}
                                        {1}{1}{1}{2}
                                        {1}{1}{1}{1}{1}

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Partitions with odd multiplicities are counted by A055922.
Odd-length multisets are counted by A000302, A027193, A058695, ranked by A026424.
Other types: A050330, A356933, A356934, A356935.
Other conditions: A001970, A006171, A007294, A089259, A107742, A356941.
A000041 counts integer partitions, strict A000009.
A001055 counts factorizations.
Cf. A000670, A011782, A055887, A072233, A117958, A270995, A304969.

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]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],OddQ[Times@@Length/@#]&]],{n,0,8}]

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(1/prod(k=1, n, (1 - x^k + O(x*x^n))^u[k])) } \\ Andrew Howroyd, Dec 30 2022

G.f.: 1/Product_{k>=1} (1 - x^k)^A027193(k). - Andrew Howroyd, Dec 30 2022

Terms a(13) and beyond from Andrew Howroyd, Dec 30 2022

A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

Original entry on oeis.org

1, 1, 5, 25, 193, 1481, 16021, 167665, 2220065, 30004273, 468585541, 7560838121, 138355144225, 2589359765305, 53501800316693, 1146089983207681, 26457132132638401, 632544682981967585, 16171678558995779845, 426926324177655018553, 11938570457328874969601

Offset: 0

Seiichi Manyama, Oct 29 2017

From Peter Bala, Nov 13 2017: (Start)
The terms of the sequence appear to be of the form 4*m + 1.
It appears that the sequence taken modulo 10 is periodic with period 5. More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
From Peter Bala, Mar 28 2022: (Start)
The above conjectures are true. See the Bala link.
a(5*n+2) == 0 (mod 5); a(5*n+3) == 0 (mod 5); a(13*n+9) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..438
Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294361 (k=1), A294362 (k=2).

Cf. A000005, A028342, A295739, A006171, A107742.

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 05 2018 *)
a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 06 2018 *)

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, numdiv(k)*x^k))))

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000005(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 07 2018

A299069 Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 11, 19, 30, 44, 69, 103, 157, 229, 341, 491, 722, 1038, 1488, 2128, 3015, 4267, 5989, 8407, 11713, 16289, 22523, 31097, 42729, 58569, 80003, 108957, 147983, 200383, 270693, 364631, 490105, 656961, 878775, 1172653, 1561626, 2074982, 2751648, 3641536, 4810009, 6341365, 8344967

Offset: 0

Ilya Gutkovskiy, Mar 09 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms

Cf. A000010, A061255, A107742, A159929, A192065, A318975.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(numtheory[phi](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 09 2018

nmax = 46; CoefficientList[Series[Product[(1 + x^k)^EulerPhi[k], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d EulerPhi[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 46}]

G.f.: Product_{k>=1} (1 + x^k)^A000010(k).

a(n) ~ exp(3^(5/3) * Zeta(3)^(1/3) * n^(2/3) / (2*Pi^(2/3))) * Zeta(3)^(1/6) / (2^(1/3) * 3^(1/6) * Pi^(5/6) * n^(2/3)). - Vaclav Kotesovec, Mar 23 2018

A280473 G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(ijk)).

Original entry on oeis.org

1, 1, 3, 6, 12, 21, 43, 70, 127, 215, 364, 591, 989, 1562, 2515, 3954, 6194, 9538, 14754, 22349, 33926, 50910, 76102, 112721, 166747, 244205, 356984, 518344, 749924, 1078711, 1547668, 2207418, 3140135, 4446572, 6276657, 8823776, 12371487, 17275879, 24061878

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. A007425, A107742, A174465, A219560, A280486.

nmax = 50; 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 = 50; A007425 = Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 1, nmax}]; s = 1 + x; Do[s *= Sum[Binomial[A007425[[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 30 2018 *)

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

A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(ij))^(ij).

Original entry on oeis.org

1, 1, 5, 11, 33, 67, 180, 366, 871, 1782, 3927, 7885, 16637, 32763, 66469, 128938, 253871, 484034, 930959, 1747304, 3292730, 6092664, 11282364, 20596790, 37568653, 67736175, 121886533, 217261372, 386216073, 681119439, 1197524035, 2091091902, 3639519280

Offset: 0

Vaclav Kotesovec, Jan 05 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. A000005, A006171, A038040, A061256, A107742, A192065, A280541.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(i*j))^(i*j), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[k*DivisorSigma[0, 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 27 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Aug 26 2018

log(a(n)) ~ (3/2)^(2/3) * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3). - Vaclav Kotesovec, Aug 28 2018

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(ij))^(ij).

Original entry on oeis.org

1, 1, 4, 10, 24, 52, 125, 253, 549, 1126, 2290, 4525, 8987, 17259, 33174, 62669, 117425, 217295, 399904, 726984, 1314257, 2354807, 4191671, 7405590, 13009916, 22696115, 39384232, 67937488, 116584833, 199001304, 338076500, 571507377, 961855945, 1611567819

Offset: 0

Vaclav Kotesovec, Jan 05 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. A000005, A006171, A038040, A061256, A107742, A192065, A280540.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j))^(i*j), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[k*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}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 27 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Aug 26 2018

Conjecture: log(a(n)) ~ 3 * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Aug 29 2018

A288414 Expansion of Product_{k>=1} (1 + x^k)^(sigma_2(k)).

Original entry on oeis.org

1, 1, 5, 15, 41, 107, 286, 700, 1735, 4162, 9803, 22673, 51822, 116376, 258548, 567197, 1230763, 2642958, 5622616, 11850537, 24769248, 51353095, 105662389, 215838649, 437890022, 882562763, 1767741732, 3519599996, 6967592060, 13717874719, 26865949075

Offset: 0

Seiichi Manyama, Jun 08 2017

nonn

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

Cf. A275585, A288389, A288419.

Product_{k>=1} (1 + x^k)^sigma_m(k): A107742 (m=0), A192065 (m=1), this sequence (m=2), A288415 (m=3).

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+q^k)^DivisorSigma(2,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

with(numtheory): seq(coeff(series(mul((1+x^k)^(sigma[2](k)),k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018

nmax = 40; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[2, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, (1+x^k)^sigma(k,2))) \\ G. C. Greubel, Oct 30 2018

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.
a(n) ~ exp(2^(5/4) * (7*Zeta(3))^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - 5^(1/4) * Pi * n^(1/4) / (2^(13/4) * 3^(7/4) * (7*Zeta(3))^(1/4))) * (7*Zeta(3))^(1/8) / (2^(15/8) * 15^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 23 2018
G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(j^2). - Ilya Gutkovskiy, Aug 26 2018

cp's OEIS Frontend

A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)*x^n = log( Sum_{n>=0} A107742(n)*x^n ).

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A192065 Expansion of Product_{k>=1} Q(x^k)^k where Q(x) = Product_{k>=1} (1 + x^k).

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A301554 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma_0(k)).

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A356932 Number of multiset partitions of integer partitions of n such that all blocks have odd size.

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A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

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A299069 Expansion of Product_{k>=1} (1 + x^k)^phi(k), where phi() is the Euler totient function (A000010).

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A280473 G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k)).

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A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(i*j).

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A109386 G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)x^n = log( Sum_{n>=0} A107742(n)x^n ).

A280473 G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(ijk)).

A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(ij))^(ij).

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(ij))^(ij).