A280486 -id:A280486 - 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

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

A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(ijk*l)).

Original entry on oeis.org

1, 1, 5, 9, 29, 49, 135, 235, 565, 995, 2177, 3821, 7900, 13728, 26974, 46606, 88128, 150644, 276283, 467647, 835708, 1400874, 2448818, 4065230, 6975307, 11470265, 19359345, 31552473, 52488142, 84808548, 139274675, 223191639, 362297234, 576064732, 925295844

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. A000041, A006171, A007426, A174465, A280486.

nmax=50; CoefficientList[Series[Product[1/(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]*(-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, Sep 08 2018 *)

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

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)).

A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(ijk*l)).

Original entry on oeis.org

1, 1, 5, 13, 47, 133, 443, 1333, 4263, 13143, 41419, 128791, 403815, 1259639, 3941579, 12310299, 38492034, 120271953, 375964616, 1174935195, 3672413322, 11477465221, 35872928244, 112117013835, 350417746650, 1095202995267, 3422999582632, 10698350241417, 33437065631262, 104505382585023

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Invert transform of A007426.

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

Cf. A000005, A007426, A011782, A129921, A280486, A280487, A304963.

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, 4)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018

nmax = 29; CoefficientList[Series[1/(1 - Sum[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 = 29; CoefficientList[Series[1/(1 - Sum[Sum[DivisorSigma[0, d] DivisorSigma[0, k/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] DivisorSigma[0, k/d], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 29}]

G.f.: 1/(1 - Sum_{k>=1} A007426(k)*x^k).

A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijkl))/(1 - x^(ijkl)).

Original entry on oeis.org

1, 2, 10, 26, 86, 210, 594, 1394, 3530, 8006, 18842, 41258, 92190, 195714, 419538, 867050, 1797568, 3625758, 7311382, 14431294, 28416514, 55010142, 106101558, 201814518, 382213566, 715473554, 1333083950, 2459265058, 4515151234, 8218572030, 14888270366, 26766878302

Offset: 0

Seiichi Manyama, Nov 01 2018

nonn

Convolution of the sequences A280486 and A280487.

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

Cf. A007426, A015128, A280486, A280487, A301554, A305050.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(   (&*[(&*[(&*[(&*[(1+x^(i*j*k*l))/(1-x^(i*j*k*l)): i in [1..m]]): j in [1..m]]): k in [1..m]]): l in [1..m]]))); // G. C. Greubel, Nov 01 2018

With[{nmax=50}, CoefficientList[Series[Product[(1 + x^(i*j*k*l))/(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]] (* G. C. Greubel, Nov 01 2018 *)

m=50; x='x+O('x^m); Vec(prod(k=1,m, ((1+x^k)/(1-x^k))^ sumdiv(k, d, numdiv(k/d)*numdiv(d)))) \\ G. C. Greubel, Nov 01 2018

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A007426(k).

A326417 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s)).

Original entry on oeis.org

1, 3, 4, 6, 4, 12, 4, 10, 10, 12, 4, 24, 4, 12, 16, 15, 4, 30, 4, 24, 16, 12, 4, 40, 10, 12, 20, 24, 4, 48, 4, 21, 16, 12, 16, 60, 4, 12, 16, 40, 4, 48, 4, 24, 40, 12, 4, 60, 10, 30, 16, 24, 4, 60, 16, 40, 16, 12, 4, 96, 4, 12, 40, 28, 16, 48, 4, 24, 16, 48, 4, 100, 4, 12, 40

Offset: 1

Ilya Gutkovskiy, Oct 18 2019

Inverse Moebius transform applied twice to A001227.

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

Cf. A000005, A001227, A007425, A007426, A133700, A280486, A318845.

Table[Sum[DivisorSigma[0, n/d] Total[Mod[Divisors[d], 2]], {d, Divisors[n]}], {n, 1, 75}]
nmax = 75; A007425 = Table[DivisorSum[n, DivisorSigma[0, #] &], {n, 1, nmax}]; Table[DivisorSum[n, A007425[[#]] &, OddQ[n/#] &], {n, 1, nmax}]
f[2, e_] := (e + 1)*(e + 2)/2; f[p_, e_] := (e + 1)*(e + 2)*(e + 3)/6; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

G.f.: Sum_{k>=1} tau_3(k) * x^k / (1 - x^(2*k)), where tau_3 = A007425.
a(n) = tau_4(n) if n odd, tau_4(n) - tau_4(n/2) if n even, where tau_4 = A007426.
a(n) = Sum_{d|n, n/d odd} tau_3(d).
a(n) = Sum_{d|n} A000005(n/d) * A001227(d).
Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A280486.
Multiplicative with a(2^e) = (e+1)*(e+2)/2, and a(p^e) = (e+1)*(e+2)*(e+3)/6 for odd primes p. - Amiram Eldar, Dec 02 2020

cp's OEIS Frontend

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

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

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A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(i*j*k*l)).

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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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A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(i*j*k*l)).

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A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(i*j*k*l))/(1 - x^(i*j*k*l)).

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A326417 Dirichlet g.f.: zeta(s)^4 * (1 - 2^(-s)).

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

A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(ijk*l)).

A304964 Expansion of 1/(1 - Sum_{i>=1, j>=1, k>=1, l>=1} x^(ijk*l)).

A321240 Expansion of Product_{i>=1, j>=1, k>=1, l>=1} (1 + x^(ijkl))/(1 - x^(ijkl)).