This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A325830 #23 May 12 2021 12:04:59
%S A325830 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 A325830 30,38,405,593,1,1574,1,11511,80,51,77,17552,1,56,92,13921,1,3060,1,
%U A325830 1439,2911,69,1,234969,56,2044,126,1998,1,46488,114,36615,137,87,1,141906
%N A325830 Number of integer partitions of 2*n having exactly 2*n submultisets.
%C A325830 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 A325830 The number of submultisets of an integer partition is the product of its multiplicities, each plus one.
%C A325830 The Heinz numbers of these partitions are given by A325793.
%H A325830 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 A325830 a(p) = 1 for prime p. - _Andrew Howroyd_, Aug 16 2019
%e A325830 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 A325830 The a(1) = 1 through a(8) = 21 partitions (A = 10, B = 11):
%e A325830 (2) (31) (411) (431) (61111) (4332) (8111111) (6532)
%e A325830 (521) (4431) (6541)
%e A325830 (5111) (5322) (7432)
%e A325830 (5331) (7531)
%e A325830 (6411) (7621)
%e A325830 (7221) (8431)
%e A325830 (7311) (8521)
%e A325830 (8211) (9421)
%e A325830 (33222) (A321)
%e A325830 (711111) (44431)
%e A325830 (53332)
%e A325830 (63331)
%e A325830 (64222)
%e A325830 (73222)
%e A325830 (76111)
%e A325830 (85111)
%e A325830 (92221)
%e A325830 (94111)
%e A325830 (A3111)
%e A325830 (B2111)
%e A325830 (91111111)
%p A325830 b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
%p A325830 `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
%p A325830 (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
%p A325830 end:
%p A325830 a:= n-> `if`(isprime(n), 1, b(2*n$3)):
%p A325830 seq(a(n), n=0..60); # _Alois P. Heinz_, Aug 16 2019
%t A325830 Table[Length[Select[IntegerPartitions[2*n],Times@@(1+Length/@Split[#])==2*n&]],{n,0,30}]
%t A325830 (* Second program: *)
%t A325830 b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
%t A325830 If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
%t A325830 Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
%t A325830 a[n_] := If[PrimeQ[n], 1, b[2n, 2n, 2n]];
%t A325830 a /@ Range[0, 60] (* _Jean-François Alcover_, May 12 2021, after _Alois P. Heinz_ *)
%o A325830 (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 A325830 Cf. A002033, A098859, A108917, A126796, A237999, A325694, A325792, A325793, A325828, A325831, A325832, A325833, A325834, A325836.
%K A325830 nonn
%O A325830 0,5
%A A325830 _Gus Wiseman_, May 25 2019
%E A325830 Terms a(31) and beyond from _Andrew Howroyd_, Aug 16 2019