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A025434 Number of partitions of n into 10 nonzero squares.

%I A025434 #25 Nov 07 2020 05:33:09
%S A025434 0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,1,0,1,1,0,1,1,0,1,2,1,1,2,1,1,2,1,2,
%T A025434 4,1,2,4,1,2,5,2,4,4,2,5,4,2,5,6,4,5,6,4,5,6,5,7,9,5,7,10,5,7,11,6,11,
%U A025434 10,6,12,10,7,13,14,10,12,14,11,12,14,12,16,19,12,16,19,12,16,22,15,21,21,15
%N A025434 Number of partitions of n into 10 nonzero squares.
%H A025434 Alois P. Heinz, <a href="/A025434/b025434.txt">Table of n, a(n) for n = 0..10000</a>
%H A025434 <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>
%F A025434 a(n) = [x^n y^10] Product_{k>=1} 1/(1 - y*x^(k^2)). - _Ilya Gutkovskiy_, Apr 19 2019
%F A025434 a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(l) * A010052(m) * A010052(o) * A010052(p) * A010052(q) *A010052(r) * A010052(n-i-j-k-l-m-o-p-q-r). - _Wesley Ivan Hurt_, Apr 19 2019
%p A025434 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
%p A025434       `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A025434       `if`(i^2>n, 0, b(n-i^2, i, t-1))))
%p A025434     end:
%p A025434 a:= n-> b(n, isqrt(n), 10):
%p A025434 seq(a(n), n=0..120);  # _Alois P. Heinz_, May 30 2014
%t A025434 b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] + If[i^2 > n, 0, b[n - i^2, i, t - 1]]]];
%t A025434 a[n_] := b[n, Sqrt[n] // Floor, 10];
%t A025434 a /@ Range[0, 120] (* _Jean-François Alcover_, Nov 07 2020, after _Alois P. Heinz_ *)
%Y A025434 Column k=10 of A243148.
%Y A025434 Cf. A010052.
%K A025434 nonn
%O A025434 0,26
%A A025434 _David W. Wilson_