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Number of partitions of n into 10 nonzero squares.
6

%I #25 Nov 07 2020 05:33:09

%S 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 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 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 Number of partitions of n into 10 nonzero squares.

%H Alois P. Heinz, <a href="/A025434/b025434.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of squares</a>

%F a(n) = [x^n y^10] Product_{k>=1} 1/(1 - y*x^(k^2)). - _Ilya Gutkovskiy_, Apr 19 2019

%F 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 b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),

%p `if`(i<1 or t<1, 0, b(n, i-1, t)+

%p `if`(i^2>n, 0, b(n-i^2, i, t-1))))

%p end:

%p a:= n-> b(n, isqrt(n), 10):

%p seq(a(n), n=0..120); # _Alois P. Heinz_, May 30 2014

%t 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 a[n_] := b[n, Sqrt[n] // Floor, 10];

%t a /@ Range[0, 120] (* _Jean-François Alcover_, Nov 07 2020, after _Alois P. Heinz_ *)

%Y Column k=10 of A243148.

%Y Cf. A010052.

%K nonn

%O 0,26

%A _David W. Wilson_