A325830 Number of integer partitions of 2*n having exactly 2*n submultisets.

0, 1, 1, 1, 3, 1, 10, 1, 21, 12, 15, 1, 121, 1, 20, 37, 309, 1, 319, 1, 309, 47, 33, 1, 3435, 30, 38, 405, 593, 1, 1574, 1, 11511, 80, 51, 77, 17552, 1, 56, 92, 13921, 1, 3060, 1, 1439, 2911, 69, 1, 234969, 56, 2044, 126, 1998, 1, 46488, 114, 36615, 137, 87, 1, 141906

Offset

0,5

Comments

If n is odd, there are no integer partitions of n with exactly n submultisets, so this sequence gives only the even-indexed terms.

The number of submultisets of an integer partition is the product of its multiplicities, each plus one.

The Heinz numbers of these partitions are given by A325793.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..700 (first 101 terms from Andrew Howroyd)

Formula

a(p) = 1 for prime p. - Andrew Howroyd, Aug 16 2019

Example

The 12 submultisets of the partition (7221) are (), (1), (2), (7), (21), (22), (71), (72), (221), (721), (722), (7221), so (7221) is counted under a(6).
The a(1) = 1 through a(8) = 21 partitions (A = 10, B = 11):
  (2)  (31)  (411)  (431)   (61111)  (4332)    (8111111)  (6532)
                    (521)            (4431)               (6541)
                    (5111)           (5322)               (7432)
                                     (5331)               (7531)
                                     (6411)               (7621)
                                     (7221)               (8431)
                                     (7311)               (8521)
                                     (8211)               (9421)
                                     (33222)              (A321)
                                     (711111)             (44431)
                                                          (53332)
                                                          (63331)
                                                          (64222)
                                                          (73222)
                                                          (76111)
                                                          (85111)
                                                          (92221)
                                                          (94111)
                                                          (A3111)
                                                          (B2111)
                                                          (91111111)

Maple

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> `if`(isprime(n), 1, b(2*n$3)):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 16 2019

Mathematica

Table[Length[Select[IntegerPartitions[2*n], Times@@(1+Length/@Split[#])==2*n&]], {n, 0, 30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
     If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
     Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := If[PrimeQ[n], 1, b[2n, 2n, 2n]];
a /@ Range[0, 60] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

Prog

(PARI) a(n)={if(n<1, 0, my(v=vector(2*n+1, k, vector(2*n))); v[1][1]=1; for(k=1, 2*n, forstep(j=#v, k, -1, for(m=1, (j-1)\k, for(i=1, 2*n\(m+1), v[j][i*(m+1)] += v[j-m*k][i])))); v[#v][2*n])} \\ Andrew Howroyd, Aug 16 2019

Crossrefs

Cf. A002033, A098859, A108917, A126796, A237999, A325694, A325792, A325793, A325828, A325831, A325832, A325833, A325834, A325836.

Sequence in context: A178866 A019427 A364850 * A331155 A008299 A016478

Adjacent sequences: A325827 A325828 A325829 * A325831 A325832 A325833

Keyword

nonn

Author

Gus Wiseman, May 25 2019

Extensions

Terms a(31) and beyond from Andrew Howroyd, Aug 16 2019

Status

approved