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

Author

Vladeta Jovovic, Jun 11 2005

Keywords

• easy
• nonn

Comments

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)

Links

• 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

Crossrefs

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.

Programs

Mathematica

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 *)

PARI

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

PARI

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 */

PARI

{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 */

Formula

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

Extensions

More terms from Paul D. Hanna, Jun 26 2005