A240944 Number of compositions of n into square parts k^2 where there are k sorts of part k^2.

1, 1, 1, 1, 3, 5, 7, 9, 15, 28, 45, 66, 99, 164, 269, 422, 651, 1028, 1654, 2637, 4149, 6522, 10350, 16467, 26091, 41205, 65174, 103339, 163833, 259361, 410376, 649827, 1029543, 1630725, 2581848, 4087797, 6473832, 10253370, 16237375, 25711316, 40714953, 64478427, 102111230, 161701086, 256062990, 405499697, 642156651

Offset: 0

Author

Joerg Arndt, Aug 04 2014

Keywords

• nonn

Links

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

Crossrefs

Cf. A006456 (compositions into squares).

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, `if`(n<0, 0,
      add(k*a(n-k^2), k=1..isqrt(n))))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 04 2014

Mathematica

a[n_] := a[n] = If[n == 0, 1, If[n<0, 0, Sum[k*a[n-k^2], {k, Sqrt[n]}]]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 29 2021, after Alois P. Heinz *)

PARI

N=66; x='x+O('x^N); Vec(1/(1 - sum(k=1, 1+sqrtint(N), k * x^(k^2))) )

Formula

G.f.: 1/(1 - sum(k>=1, k * x^(k^2)) ).