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A332233 Number of integer partitions lambda (of any k) satisfying n = max_{p:lambda} p*m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.

%I A332233 #24 Oct 30 2023 18:29:17
%S A332233 1,1,4,10,44,84,528,864,4944,12720,56832,89856,882432,1209600,6036480,
%T A332233 20017152,98592768,141834240,1202135040,1625702400,12997877760,
%U A332233 35291013120,124429271040,191102976000,2350327726080,4064999178240,15972386734080,47163577466880
%N A332233 Number of integer partitions lambda (of any k) satisfying n = max_{p:lambda} p*m(p,lambda), where m(p,lambda) is the multiplicity of part p in lambda.
%C A332233 a(0) = 1 by convention.
%H A332233 Alois P. Heinz, <a href="/A332233/b332233.txt">Table of n, a(n) for n = 0..1000</a>
%H A332233 Wikipedia, <a href="https://www.wikipedia.org/wiki/integer_partition">Integer Partition</a>
%F A332233 a(n) = Sum_{k=n..A024916(n)} A134979(k,n) for n > 0.
%F A332233 a(n) = A131385(n+1) - A131385(n), for n>0. - _Ridouane Oudra_, Oct 30 2023
%e A332233 a(2) = 4: 2, 11, 21, 211.
%e A332233 a(3) = 10: 3, 31, 32, 111, 311, 321, 2111, 3111, 3211, 32111.
%e A332233 a(4) = 44: 4, 22, 41, 42, 43, 221, 322, 411, 421, 422, 431, 432, 1111, 2211, 3221, 4111, 4211, 4221, 4311, 4321, 4322, 21111, 22111, 31111, 32211, 41111, 42111, 42211, 43111, 43211, 43221, 221111, 321111, 322111, 421111, 422111, 431111, 432111, 432211, 3221111, 4221111, 4321111, 4322111, 43221111.
%p A332233 b:= proc(n, i, m, t) option remember; `if`(n=0, m,
%p A332233       `if`(i<1 or m=0 and n<t, 0, add(b(n-i*j, i-1,
%p A332233            `if`(t=i*j, 1, m), t), j=0..min(t, n)/i)))
%p A332233     end:
%p A332233 a:= proc(k) option remember; local r, n, t; r:=0;
%p A332233       for n from k do t:= b(n$2, 0, k);
%p A332233       if t=0 then break else r:=r+t fi od; r
%p A332233     end: a(0):=1:
%p A332233 seq(a(n), n=0..20);
%t A332233 $RecursionLimit = 2000;
%t A332233 b[n_, i_, m_, t_] := b[n, i, m, t] = If[n==0, m, If[i<1 || m==0 && n<t, 0, Sum[b[n - i j, i-1, If[t == i j, 1, m], t], {j, 0, Min[t, n]/i}]]];
%t A332233 a[0] = 1;
%t A332233 a[k_] := a[k] = Module[{r = 0, n, t}, For[n = k, True, n++, t = b[n, n, 0, k]; If[t == 0, Break[], r += t]]; r];
%t A332233 a /@ Range[0, 27] (* _Jean-François Alcover_, May 08 2020, after Maple *)
%Y A332233 Column sums of A134979.
%Y A332233 Cf. A024916.
%Y A332233 Cf. A131385 (partial sum).
%K A332233 nonn
%O A332233 0,3
%A A332233 _Alois P. Heinz_, Feb 07 2020