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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.

%I A111212 #25 Jun 02 2022 09:43:41
%S A111212 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,
%T A111212 101,110,127,144,134,172,180,189,205,235,237,266,282,303,323,352,346,
%U A111212 391,403,436,453,497,492,547,555,596,606,661,670,724,741,775,806,861
%N 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.
%H A111212 Alois P. Heinz, <a href="/A111212/b111212.txt">Table of n, a(n) for n = 0..500</a> (first 301 terms from Joerg Arndt)
%e A111212 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
%p A111212 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);
%p A111212 # second Maple program:
%p A111212 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, {}, `if`(n=0, {0},
%p A111212       {b(n, i-1)[], map(x->x+i^2, b(n-i, min(n-i, i-1)))[]}))
%p A111212     end:
%p A111212 a:= n-> nops(b(n$2)):
%p A111212 seq(a(n), n=0..65); # _Alois P. Heinz_, Apr 18 2019
%t A111212 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 *)
%t A111212 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 *)
%Y A111212 Cf. A000009, A069999.
%K A111212 easy,nonn
%O A111212 0,4
%A A111212 _Vladeta Jovovic_, Oct 25 2005
%E A111212 Corrected term a(2), _Joerg Arndt_, Apr 18 2019