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A025443 Number of partitions of n into 4 distinct nonzero squares.

%I A025443 #30 Jun 09 2024 15:48:27
%S A025443 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%T A025443 0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,1,0,0,1,0,0,1,0,0,0,0,1,1,0,1,1,0,
%U A025443 0,0,1,1,0,0,1,1,0,0,3,1,0,1,0,0,1,1,1,1,0,0,2,1,0,1,2,2,0,0,1,2,0,0,3,0,0,2,1,1
%N A025443 Number of partitions of n into 4 distinct nonzero squares.
%H A025443 Alois P. Heinz, <a href="/A025443/b025443.txt">Table of n, a(n) for n = 0..20000</a>
%H A025443 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A025443 a(n) = [x^n y^4] Product_{k>=1} (1 + y*x^(k^2)). - _Ilya Gutkovskiy_, Apr 22 2019
%p A025443 b:= proc(n,i,t) option remember; `if`(n=0, `if`(t=0,1,0),
%p A025443       `if`(t*i^2<n, 0, `if`(i=1, 0, b(n,i-1,t))+
%p A025443       `if`(i^2>n, 0, b(n-i^2,i-1,t-1))))
%p A025443     end:
%p A025443 a:= n-> b(n, isqrt(n), 4):
%p A025443 seq(a(n), n=0..150);  # _Alois P. Heinz_, Feb 07 2013
%t A025443 b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t==0, 1, 0], If[t*i^2<n, 0, If[i == 1, 0, b[n, i-1, t]] + If[i^2>n, 0, b[n-i^2, i-1, t-1]]]]; a[n_] := b[n, Sqrt[n] // Floor, 4]; Table[a[n], {n, 0, 150}] (* _Jean-François Alcover_, Feb 29 2016, after _Alois P. Heinz_*)
%t A025443 dnzs[n_]:=Length[Select[IntegerPartitions[n,{4}],Length[Union[#]]==4&&AllTrue[ Sqrt[ #], IntegerQ] && FreeQ[#,0]&]]; Array[dnzs,110,0] (* _Harvey P. Dale_, Jun 09 2024 *)
%Y A025443 Cf. A025428 (not necessarily distinct), A025376-A025394 (subsequences), A025417 (greedy inverse).
%Y A025443 Column k=4 of A341040.
%K A025443 nonn,look
%O A025443 0,79
%A A025443 _David W. Wilson_