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A318949 Number of ways to write n as an orderless product of orderless sums. 27
1, 2, 3, 8, 7, 17, 15, 36, 36, 56, 56, 123, 101, 165, 197, 310, 297, 490, 490, 767, 837, 1114, 1255, 1925, 1986, 2638, 3110, 4108, 4565, 6201, 6842, 9043, 10311, 12904, 14988, 19398, 21637, 26995, 31488, 39180, 44583, 55418, 63261, 77627, 89914, 108068, 124754 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..10000

FORMULA

Dirichlet g.f.: Product_{k>=2} 1 / (1 - k^(-s))^p(k), where p(k) = number of partitions of k (A000041). - Ilya Gutkovskiy, Oct 26 2019

EXAMPLE

The a(6) = 17 ways:

  (6)              (2)*(3)

  (3+3)            (2)*(2+1)

  (4+2)            (2)*(1+1+1)

  (5+1)            (1+1)*(3)

  (2+2+2)          (1+1)*(2+1)

  (3+2+1)          (1+1)*(1+1+1)

  (4+1+1)

  (2+2+1+1)

  (3+1+1+1)

  (2+1+1+1+1)

  (1+1+1+1+1+1)

MATHEMATICA

facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]];

prodsums[n_]:=Union[Sort/@Join@@Table[Tuples[IntegerPartitions/@fac], {fac, facs[n]}]];

Table[Length[prodsums[n]], {n, 30}]

PROG

(PARI) MultEulerT(u)={my(v=vector(#u)); v[1]=1; for(k=2, #u, forstep(j=#v\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j]+=binomial(e+u[k]-1, e)*v[i]))); v}

seq(n)={MultEulerT(vector(n, n, numbpart(n)))} \\ Andrew Howroyd, Oct 26 2019

CROSSREFS

Cf. A000041, A001055, A001970, A063834, A065026, A066739, A066815, A281113, A284639, A318948.

Sequence in context: A196828 A171046 A250116 * A100836 A173162 A198104

Adjacent sequences:  A318946 A318947 A318948 * A318950 A318951 A318952

KEYWORD

nonn

AUTHOR

Gus Wiseman, Sep 05 2018

STATUS

approved

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)