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Number of integer partitions of 2*n having exactly 2*n submultisets.
11

%I #23 May 12 2021 12:04:59

%S 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,

%T 30,38,405,593,1,1574,1,11511,80,51,77,17552,1,56,92,13921,1,3060,1,

%U 1439,2911,69,1,234969,56,2044,126,1998,1,46488,114,36615,137,87,1,141906

%N Number of integer partitions of 2*n having exactly 2*n submultisets.

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

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

%C The Heinz numbers of these partitions are given by A325793.

%H Alois P. Heinz, <a href="/A325830/b325830.txt">Table of n, a(n) for n = 0..700</a> (first 101 terms from Andrew Howroyd)

%F a(p) = 1 for prime p. - _Andrew Howroyd_, Aug 16 2019

%e 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).

%e The a(1) = 1 through a(8) = 21 partitions (A = 10, B = 11):

%e (2) (31) (411) (431) (61111) (4332) (8111111) (6532)

%e (521) (4431) (6541)

%e (5111) (5322) (7432)

%e (5331) (7531)

%e (6411) (7621)

%e (7221) (8431)

%e (7311) (8521)

%e (8211) (9421)

%e (33222) (A321)

%e (711111) (44431)

%e (53332)

%e (63331)

%e (64222)

%e (73222)

%e (76111)

%e (85111)

%e (92221)

%e (94111)

%e (A3111)

%e (B2111)

%e (91111111)

%p b:= proc(n, i, p) option remember; `if`(n=0 or i=1,

%p `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,

%p (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))

%p end:

%p a:= n-> `if`(isprime(n), 1, b(2*n$3)):

%p seq(a(n), n=0..60); # _Alois P. Heinz_, Aug 16 2019

%t Table[Length[Select[IntegerPartitions[2*n],Times@@(1+Length/@Split[#])==2*n&]],{n,0,30}]

%t (* Second program: *)

%t b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,

%t If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);

%t Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];

%t a[n_] := If[PrimeQ[n], 1, b[2n, 2n, 2n]];

%t a /@ Range[0, 60] (* _Jean-François Alcover_, May 12 2021, after _Alois P. Heinz_ *)

%o (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

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

%K nonn

%O 0,5

%A _Gus Wiseman_, May 25 2019

%E Terms a(31) and beyond from _Andrew Howroyd_, Aug 16 2019