A052337 Number of partitions into at most a(1) copies of 1, a(2) copies of 2, ...

1, 1, 1, 2, 2, 3, 5, 6, 8, 10, 13, 17, 21, 27, 34, 42, 53, 65, 80, 98, 119, 146, 177, 213, 258, 309, 370, 443, 528, 628, 746, 883, 1044, 1231, 1449, 1703, 1997, 2338, 2734, 3190, 3718, 4325, 5025, 5830, 6754, 7816, 9032, 10422, 12016, 13832, 15907, 18274

Offset: 0

Author

Christian G. Bower, Dec 19 1999

Keywords

• nonn
• eigen

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A289501.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=0..min(n/i, a(i)))))
    end:
a:= n-> `if`(n=0, 0, 1)+b(n, n-1):
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 12 2018

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1], {j, 0, Min[n/i, a[i]]}]]];
a[n_] := If[n == 0, 0, 1] + b[n, n - 1];
a /@ Range[0, 70] (* Jean-François Alcover, Nov 23 2020, after Alois P. Heinz *)

Formula

E.g.f. satisfies A(x) = Product_{i>=1} (1-x^(a(i)*(i+1)))/(1-x^i).