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A107742
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G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).
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57
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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
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OFFSET
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0,3
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COMMENTS
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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}}
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}}
The non-constant block version is A261049.
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)
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LINKS
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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A072233 counts partitions by sum and length.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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