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A025433 Number of partitions of n into 9 nonzero squares.

%I A025433 #19 Apr 19 2019 18:36:00
%S A025433 0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,1,0,1,1,0,1,2,1,1,2,1,1,2,1,2,4,
%T A025433 1,2,4,1,2,4,2,4,4,2,4,4,2,5,6,3,5,5,4,5,5,5,6,9,5,6,9,4,7,10,5,10,9,
%U A025433 6,11,9,6,11,13,9,11,12,9,11,13,11,14,16,11,14,16,10,13,20,13,18,19,12,20,18,13
%N A025433 Number of partitions of n into 9 nonzero squares.
%H A025433 Alois P. Heinz, <a href="/A025433/b025433.txt">Table of n, a(n) for n = 0..10000</a>
%H A025433 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A025433 a(n) = [x^n y^9] Product_{k>=1} 1/(1 - y*x^(k^2)). - _Ilya Gutkovskiy_, Apr 19 2019
%F A025433 a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/6)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(l) * A010052(m) * A010052(o) * A010052(p) * A010052(q) * A010052(n-i-j-k-l-m-o-p-q). - _Wesley Ivan Hurt_, Apr 19 2019
%p A025433 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
%p A025433       `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A025433       `if`(i^2>n, 0, b(n-i^2, i, t-1))))
%p A025433     end:
%p A025433 a:= n-> b(n, isqrt(n), 9):
%p A025433 seq(a(n), n=0..120);  # _Alois P. Heinz_, May 30 2014
%t A025433 a[n_] := Count[ PowersRepresentations[n, 9, 2], pr_List /; FreeQ[pr, 0]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Dec 27 2012 *)
%Y A025433 Column k=9 of A243148.
%K A025433 nonn
%O A025433 0,25
%A A025433 _David W. Wilson_