A107742 - OEIS

%I #68 Sep 30 2024 03:24:21

%S 1,1,2,4,6,10,17,25,38,59,86,125,184,260,369,524,726,1005,1391,1894,

%T 2576,3493,4687,6272,8373,11090,14647,19294,25265,32991,42974,55705,

%U 72025,92895,119349,152965,195592,249280,316991,402215,508932,642598,809739,1017850,1276959,1599015,1997943,2491874,3102477,3855165,4782408,5922954

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

%C From _Gus Wiseman_, Sep 13 2022: (Start)

%C 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:

%C   {{1}}  {{2}}      {{3}}          {{4}}

%C          {{1},{1}}  {{1,2}}        {{1},{3}}

%C                     {{1},{2}}      {{2},{2}}

%C                     {{1},{1},{1}}  {{1},{1,2}}

%C                                    {{1},{1},{2}}

%C                                    {{1},{1},{1},{1}}

%C Intervals are counted by A001227, ranked by A073485.

%C The initial version is A007294.

%C The strict version is A327731.

%C The version for gapless multisets instead of intervals is A356941.

%C The case of strict partitions is A356957.

%C 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:

%C   {{1}}  {{2}}    {{3}}        {{4}}

%C          {{1,1}}  {{1,1,1}}    {{2,2}}

%C                   {{1},{2}}    {{1},{3}}

%C                   {{1},{1,1}}  {{1,1,1,1}}

%C                                {{2},{1,1}}

%C                                {{1},{1,1,1}}

%C Constant multisets are counted by A000005, ranked by A000961.

%C The non-strict version is A006171.

%C The unlabeled version is A089259.

%C The non-constant block version is A261049.

%C The version for twice-partitions is A279786, factorizations A296131.

%C 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:

%C   {{1}}  {{2}}      {{3}}          {{4}}

%C          {{1},{1}}  {{1,1,1}}      {{1},{3}}

%C                     {{1},{2}}      {{2},{2}}

%C                     {{1},{1},{1}}  {{1},{1,1,1}}

%C                                    {{1},{1},{2}}

%C                                    {{1},{1},{1},{1}}

%C The strict version is A327731 (also).

%C (End)

%H Vaclav Kotesovec, <a href="/A107742/b107742.txt">Table of n, a(n) for n = 0..10000</a>

%H Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, <a href="https://doi.org/10.1007/s44007-024-00134-w">A unified treatment of families of partition functions</a>, La Matematica (2024). Preprint available as <a href="https://arxiv.org/abs/2303.02240">arXiv:2303.02240</a> [math.CO], 2023.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Euler transform of A001227.

%F Weigh transform of A000005.

%F 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

%F G.f.: A(x) = exp( Sum_{n>=1} sigma(n)*x^n/(1-x^(2n)) /n ). - _Paul D. Hanna_, Mar 28 2009

%F G.f.: Product_{n>=1} Q(x^n) where Q(x) is the g.f. of A000009. - _Joerg Arndt_, Feb 27 2014

%F a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A109386(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Jun 04 2017

%F Conjecture: log(a(n)) ~ Pi*sqrt(n*log(n)/6). - _Vaclav Kotesovec_, Aug 29 2018

%t 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 *)

%t nmax = 50; CoefficientList[Series[Product[(1+x^k)^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Mar 23 2018 *)

%t 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 *)

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};

%t Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@chQ/@#&]],{n,0,5}] (* _Gus Wiseman_, Sep 13 2022 *)

%o (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_ */

%o (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 */

%o (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 */

%Y Cf. A006171, A109386, A219554, A280473, A280486, A288007.

%Y 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).

%Y A000041 counts integer partitions, strict A000009.

%Y A000110 counts set partitions.

%Y A072233 counts partitions by sum and length.

%Y Cf. A001055, A001970, A011782, A055887, A061260, A270995, A279784, A294617, A304969.

%K easy,nonn

%O 0,3

%A _Vladeta Jovovic_, Jun 11 2005

%E More terms from _Paul D. Hanna_, Jun 26 2005