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A240944 Number of compositions of n into square parts k^2 where there are k sorts of part k^2.

%I A240944 #17 Aug 29 2021 02:08:01
%S A240944 1,1,1,1,3,5,7,9,15,28,45,66,99,164,269,422,651,1028,1654,2637,4149,
%T A240944 6522,10350,16467,26091,41205,65174,103339,163833,259361,410376,
%U A240944 649827,1029543,1630725,2581848,4087797,6473832,10253370,16237375,25711316,40714953,64478427,102111230,161701086,256062990,405499697,642156651
%N A240944 Number of compositions of n into square parts k^2 where there are k sorts of part k^2.
%H A240944 Alois P. Heinz, <a href="/A240944/b240944.txt">Table of n, a(n) for n = 0..1000</a>
%F A240944 G.f.: 1/(1 - sum(k>=1, k * x^(k^2)) ).
%p A240944 a:= proc(n) option remember; `if`(n=0, 1, `if`(n<0, 0,
%p A240944       add(k*a(n-k^2), k=1..isqrt(n))))
%p A240944     end:
%p A240944 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 04 2014
%t A240944 a[n_] := a[n] = If[n == 0, 1, If[n<0, 0, Sum[k*a[n-k^2], {k, Sqrt[n]}]]];
%t A240944 Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Aug 29 2021, after _Alois P. Heinz_ *)
%o A240944 (PARI) N=66; x='x+O('x^N); Vec(1/(1 - sum(k=1, 1+sqrtint(N), k * x^(k^2))) )
%Y A240944 Cf. A006456 (compositions into squares).
%K A240944 nonn
%O A240944 0,5
%A A240944 _Joerg Arndt_, Aug 04 2014