A288007 -id:A288007 - 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

A320908 Expansion of Product_{k>=1} theta_4(x^k), where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, -2, -2, 2, 4, 6, -6, -2, -8, -12, 2, 6, 20, 14, 22, -2, -14, -34, -20, -42, -48, 34, 10, 50, 48, 80, 82, 52, -16, -30, -142, -130, -138, -226, -54, -70, 80, 190, 310, 238, 392, 178, 178, 86, -40, -148, -582, -506, -546, -680, -656, -126, -336, 262, 428, 930

Offset: 0

Ilya Gutkovskiy, Oct 23 2018

Convolution of A288007 and A288098.

Convolution inverse of A301554.

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

Cf. A000005, A000203, A002448, A288007, A288098, A301554, A320067, A320068.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(&*[(1-x^(j*k))/(1+x^(j*k)):j in [1..2*m]]): k in [1..2*m]]) )); // G. C. Greubel, Oct 29 2018

with(numtheory): seq(coeff(series(mul(((1-x^k)/(1+x^k))^tau(k),k=1..n),x,n+1), x, n), n = 0 .. 60); # Muniru A Asiru, Oct 23 2018

nmax = 55; CoefficientList[Series[Product[EllipticTheta[4, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[Product[((1 - x^k)/(1 + x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 55; CoefficientList[Series[Exp[-Sum[DivisorSigma[1, k] x^k (2 + x^k)/(k (1 - x^(2 k))), {k, 1, nmax}]], {x, 0, nmax}], x]

N=99; x='x+O('x^N); Vec(prod(k=1, N, ((1-x^k)/(1+x^k))^numdiv(k))) \\ Seiichi Manyama, Oct 25 2018

G.f.: Product_{i>=1, j>=1} (1 - x^(i*j))/(1 + x^(i*j)).
G.f.: Product_{k>=1} ((1 - x^k)/(1 + x^k))^d(k), where d(k) is the number of divisors of k (A000005).
G.f.: exp(-Sum_{k>=1} sigma(k)*x^k*(2 + x^k)/(k*(1 - x^(2*k)))).

A288421 Expansion of Product_{k>=1} 1/(1 + x^k)^sigma(k).

Original entry on oeis.org

1, -1, -2, -2, 1, 5, 4, 10, 6, -5, -20, -27, -37, -32, -18, 23, 82, 128, 190, 185, 143, 43, -160, -424, -662, -968, -1058, -971, -571, 238, 1326, 2748, 4195, 5301, 5930, 5473, 3353, 55, -5346, -12106, -19421, -26603, -31950, -33248, -29344, -17469, 2343, 30966

Offset: 0

Seiichi Manyama, Jun 09 2017

sign

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

Cf. A192065, A288418.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), this sequence (m=1), A288422 (m=2), A288423 (m=3).

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

nmax = 50; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[1, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 09 2017 *)

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

Convolution inverse of A192065.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288418(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

A288422 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_2(k)).

Original entry on oeis.org

1, -1, -4, -6, 0, 24, 51, 89, 47, -152, -578, -1149, -1482, -738, 2384, 8901, 18476, 26774, 24151, -7143, -86804, -226605, -406442, -539872, -441822, 181268, 1671148, 4240334, 7618777, 10551340, 10218856, 1973258, -20190349, -61492391, -121880826

Offset: 0

Seiichi Manyama, Jun 09 2017

sign

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

Cf. A288414, A288419.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), this sequence (m=2), A288423 (m=3).

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

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

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

Convolution inverse of A288414.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288419(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

A288423 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_3(k)).

Original entry on oeis.org

1, -1, -8, -20, -8, 134, 512, 1062, 406, -5319, -22532, -51843, -58869, 83035, 648412, 1947384, 3665081, 3040131, -8272126, -46481039, -128400098, -234847560, -215189896, 378947363, 2437661943, 7036096665, 13868464378, 16886982518, -4042283985, -93095770772

Offset: 0

Seiichi Manyama, Jun 09 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..7039

Cf. A288415, A288420.

Product_{k>=1} 1/(1 + x^k)^sigma_m(k): A288007 (m=0), A288421 (m=1), A288422 (m=2), this sequence (m=3).

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1+q^k)^DivisorSigma(3,k): k in [1..m]]) )); // G. C. Greubel, Oct 30 2018

with(numtheory): seq(coeff(series(mul(1/(1+x^k)^(sigma[3](k)),k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 31 2018

nmax = 40; CoefficientList[Series[Product[1/(1+x^k)^DivisorSigma[3, 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/(1+x^k)^sigma(k,3))) \\ G. C. Greubel, Oct 30 2018

Convolution inverse of A288415.
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A288420(k)*a(n-k) for n > 0.
G.f.: exp(-Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Oct 29 2018

A320245 G.f.: Product_{k>=1, j>=1} 1/((1 + x^(kj)) (1 - x^(k*j))^2).

Original entry on oeis.org

1, 1, 4, 6, 17, 25, 59, 89, 187, 284, 545, 828, 1505, 2270, 3930, 5904, 9861, 14695, 23827, 35248, 55775, 81882, 126874, 184870, 281467, 407065, 610193, 876282, 1295892, 1848144, 2700398, 3825912, 5530337, 7786022, 11145541, 15597196, 22131170, 30792303

Offset: 0

Vaclav Kotesovec, Oct 08 2018

nonn

Convolution of A288007 and A320236.

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

Cf. A158441, A320236, A320238.

nmax = 50; CoefficientList[Series[Product[1/((1+x^(k*j))*(1-x^(k*j))^2), {k, 1, nmax}, {j, 1, Floor[nmax/k]+1}], {x, 0, nmax}], x]

A327852 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A092869.

Original entry on oeis.org

1, -1, -1, 1, -1, 1, 1, -3, 1, 2, 0, 2, -2, -2, -1, 3, 1, -5, 2, 0, 0, 8, -4, -7, 5, -2, 0, 1, -8, 0, 12, 2, -3, -1, -7, 9, 4, -7, -7, -6, 10, 9, 2, -6, -14, 15, 3, -15, 19, -30, 6, 37, -31, 10, 9, -23, 20, 4, -29, 4, 14, 4, -13, 23, -14, -19, 39, -29, -23, 35, 0, -34, 48

Offset: 0

Seiichi Manyama, Sep 28 2019

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Product_{k>=1} (1 - x^k)^(Sum_{d|k} (b/d)), where (m/n) is the Kronecker symbol: this sequence (b=2), A288007 (b=4), A327688 (b=5).

Cf. A035185, A092869, A327851.

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(2, d))))

G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(8*j-1))) * (1-x^(i*(8*j-7))) / ((1-x^(i*(8*j-3))) * (1-x^(i*(8*j-5)))).

G.f.: Product_{k>=1} (1-x^k)^A035185(k).

A327746 Expansion of Product_{i>=1, j>=1} 1 / (1 + (-x)^(i(2j - 1))).

Original entry on oeis.org

1, 1, 0, 2, 2, 2, 3, 3, 6, 7, 8, 9, 14, 16, 17, 26, 30, 35, 43, 52, 62, 77, 87, 104, 133, 152, 173, 212, 251, 287, 344, 397, 465, 549, 627, 729, 864, 986, 1127, 1325, 1524, 1740, 2009, 2306, 2641, 3047, 3455, 3942, 4549, 5157, 5846, 6700, 7605, 8608

Offset: 0

Ilya Gutkovskiy, Sep 23 2019

nonn

Jason Bard, Table of n, a(n) for n = 0..2000

Cf. A001227, A107742, A288007, A327731.

nmax = 53; CoefficientList[Series[Product[1/(1 + (-x)^k)^DivisorSum[k, Mod[#, 2] &], {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[(-1)^k Sum[(-1)^(k/d) d DivisorSum[d, Mod[#, 2] &], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 53}]

G.f.: Product_{k>=1} 1 / (1 + (-x)^k)^A001227(k).

cp's OEIS Frontend

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

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A320908 Expansion of Product_{k>=1} theta_4(x^k), where theta_4() is the Jacobi theta function.

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A288421 Expansion of Product_{k>=1} 1/(1 + x^k)^sigma(k).

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A288422 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_2(k)).

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A288423 Expansion of Product_{k>=1} 1/(1 + x^k)^(sigma_3(k)).

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A320245 G.f.: Product_{k>=1, j>=1} 1/((1 + x^(k*j)) * (1 - x^(k*j))^2).

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A327852 Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A092869.

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A320245 G.f.: Product_{k>=1, j>=1} 1/((1 + x^(kj)) (1 - x^(k*j))^2).

A327746 Expansion of Product_{i>=1, j>=1} 1 / (1 + (-x)^(i(2j - 1))).