A111212 Number of distinct integers d(pi), where pi ranges over all partitions of n into distinct parts and d(pi) = sum of squares of parts of pi.

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 10, 12, 12, 18, 20, 23, 27, 35, 32, 46, 48, 55, 59, 79, 74, 94, 101, 110, 127, 144, 134, 172, 180, 189, 205, 235, 237, 266, 282, 303, 323, 352, 346, 391, 403, 436, 453, 497, 492, 547, 555, 596, 606, 661, 670, 724, 741, 775, 806, 861

Offset: 0

Author

Vladeta Jovovic, Oct 25 2005

Keywords

• easy
• nonn

Examples

			The 8 partitions of 9 into distinct parts have these sums of squares:  81, 65, 53, 45, 41, 41, 35, 29, where 41 = 6^2 + 2^2 + 1^2 = 5^2 + 4^2, so that a(9) = 7. - _Clark Kimberling_, Apr 13 2014
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500 (first 301 terms from Joerg Arndt)

Crossrefs

Cf. A000009, A069999.

Programs

Maple

seq(`if`(m=2, 1, nops(simplify(coeff(series(mul(1+x^(k^2)*y^k, k=1..61), y, 61), y, m)))), m=0..60);
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2x+i^2, b(n-i, min(n-i, i-1)))[]}))
    end:
a:= n-> nops(b(n$2)):
seq(a(n), n=0..65); # Alois P. Heinz, Apr 18 2019

Mathematica

z = 40; g[n_] := n^2; q[n_] := q[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; Map[Length, Map[Union, Table[Total[Map[g, q[n][[k]]]], {n, 1, z}, {k, 1, PartitionsQ[n]}]]] (* Clark Kimberling, Apr 13 2014 *)
terms = 60; s = (Product[1+x^k^2*y^k, {k, terms}] + O[y]^terms) + O[x]^terms^2; Join[{1, 1}, Length /@ CoefficientList[s, y][[3 ;; terms]]] (* Jean-François Alcover, Jan 29 2018, adapted from Maple *)

Extensions

Corrected term a(2), Joerg Arndt, Apr 18 2019