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

0,4

Links

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

Example

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

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)/2<n, {}, `if`(n=0, {0},
      {b(n, i-1)[], map(x->x+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 *)

Crossrefs

Cf. A000009, A069999.

Sequence in context: A301513 A066639 A370808 * A338317 A141286 A165686

Adjacent sequences: A111209 A111210 A111211 * A111213 A111214 A111215

Keyword

easy,nonn

Author

Vladeta Jovovic, Oct 25 2005

Extensions

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

Status

approved