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

Author

Alois P. Heinz, Feb 07 2020

Keywords

• nonn

Comments

a(0) = 1 by convention.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Wikipedia, Integer Partition

Crossrefs

Column sums of A134979.

Cf. A024916.

Cf. A131385 (partial sum).

Programs

Maple

b:= proc(n, i, m, t) option remember; `if`(n=0, m,
      `if`(i<1 or m=0 and n
				

Mathematica

$RecursionLimit = 2000;
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n==0, m, If[i<1 || m==0 && nJean-François Alcover, May 08 2020, after Maple *)

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