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