A188216 Expansion of 1 + Sum_{n>=1} (x^(n^2) / Product_{k>=n} (1 - x^k)).

1, 1, 1, 2, 4, 5, 8, 12, 17, 25, 34, 46, 64, 86, 114, 151, 200, 258, 335, 431, 552, 703, 891, 1121, 1411, 1764, 2196, 2725, 3374, 4155, 5111, 6260, 7650, 9319, 11329, 13726, 16608, 20031, 24114, 28962, 34725, 41529, 49595, 59095, 70304, 83476, 98968, 117109

Offset: 0

Author

Joerg Arndt, Mar 24 2011

Keywords

• nonn

Comments

Number of partitions of n such that if k is the least part, then k occurs at least k times. - Joerg Arndt, Apr 17 2011

Links

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

Crossrefs

Cf. A096403 (expansion of sum(n>=1, x^(n^2) / prod(k>=n+1, 1-x^k)) ).

Cf. A003114 (largest part k occurs at least k times).

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> `if`(n=0, 1, add(b(n-j^2, j), j=1..isqrt(n))):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 03 2021

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := If[n==0, 1, Sum[b[n - j^2, j], {j, 1, Sqrt[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, Jan 25 2021, after Alois P. Heinz *)

PARI

N=55; x='x+O('x^N);
t=1+sum(n=1,N,x^(n^2)/prod(k=n,N,1-x^k));
Vec(t)