A073336 Total number of square parts in all partitions of n.

0, 1, 2, 4, 8, 13, 21, 33, 51, 76, 111, 159, 226, 315, 435, 593, 805, 1077, 1435, 1893, 2486, 3237, 4198, 5405, 6935, 8843, 11235, 14201, 17893, 22437, 28052, 34929, 43371, 53653, 66201, 81410, 99876, 122155, 149063, 181399, 220280, 266811, 322524, 388960

Offset: 0

Author

Vladeta Jovovic, Aug 22 2002

Keywords

• easy
• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Crossrefs

Cf. A000041, A046951, A309535.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0, 0], add((l->l+[0, `if`(j>0 and issqr(i),
       l[1]*j, 0)])(b(n-i*j, i-1)), j=0..iquo(n, i))))
    end:
a:= n-> b(n, n)[2] :
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 19 2013

Mathematica

b[n_, i_] :=  b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[{l}, l+{0, If[j>0 && IntegerQ[Sqrt[i]], l[[1]]*j, 0]}][b[n-i*j, i-1]], {j, 0, Quotient[n, i]}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 13 2015, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} A046951(k)*A000041(n-k).

G.f.: Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Jan 24 2017

Extensions

More terms from Emeric Deutsch, Nov 18 2004

a(0) inserted by Alois P. Heinz, Feb 19 2013