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.

1, 1, 4, 10, 44, 84, 528, 864, 4944, 12720, 56832, 89856, 882432, 1209600, 6036480, 20017152, 98592768, 141834240, 1202135040, 1625702400, 12997877760, 35291013120, 124429271040, 191102976000, 2350327726080, 4064999178240, 15972386734080, 47163577466880

Offset

0,3

Comments

a(0) = 1 by convention.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Wikipedia, Integer Partition

Formula

a(n) = Sum_{k=n..A024916(n)} A134979(k,n) for n > 0.

a(n) = A131385(n+1) - A131385(n), for n>0. - Ridouane Oudra, Oct 30 2023

Example

a(2) = 4: 2, 11, 21, 211.
a(3) = 10: 3, 31, 32, 111, 311, 321, 2111, 3111, 3211, 32111.
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.

Maple

b:= proc(n, i, m, t) option remember; `if`(n=0, m,
      `if`(i<1 or m=0 and n<t, 0, add(b(n-i*j, i-1,
           `if`(t=i*j, 1, m), t), j=0..min(t, n)/i)))
    end:
a:= proc(k) option remember; local r, n, t; r:=0;
      for n from k do t:= b(n$2, 0, k);
      if t=0 then break else r:=r+t fi od; r
    end: a(0):=1:
seq(a(n), n=0..20);

Mathematica

$RecursionLimit = 2000;
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}]]];
a[0] = 1;
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];
a /@ Range[0, 27] (* Jean-François Alcover, May 08 2020, after Maple *)

Crossrefs

Column sums of A134979.

Cf. A024916.

Cf. A131385 (partial sum).

Sequence in context: A194993 A280907 A372648 * A149220 A149221 A149222

Adjacent sequences: A332230 A332231 A332232 * A332234 A332235 A332236

Keyword

nonn

Author

Alois P. Heinz, Feb 07 2020

Status

approved