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A093115 Number of partitions of n^2 into squares not greater than n.

%I A093115 #14 Dec 29 2016 16:08:55
%S A093115 1,1,1,1,5,7,10,13,17,108,159,228,317,430,572,748,5753,8125,11266,
%T A093115 15376,20672,27430,35942,46575,59717,523905,708028,946875,1253880,
%U A093115 1645224,2140099,2761318,3535658,4494602,5674753,7118724,69766770,90940578,117756370
%N A093115 Number of partitions of n^2 into squares not greater than n.
%H A093115 Alois P. Heinz, <a href="/A093115/b093115.txt">Table of n, a(n) for n = 0..1000</a>
%F A093115 Coefficient of x^(n^2) in the series expansion of Product_{k=1..floor(sqrt(n))} 1/(1 - x^(k^2)). - _Vladeta Jovovic_, Mar 24 2004
%e A093115 n=6: 6^2 = 9*2^2 = 8*2^2+4*1^2 = 7*2^2+8*1^2 = 6*2^2+12*1^2 = 5*2^2+16*1^2 = 4*2^2+20*1^2 = 3*2^2+24*1^2 = 2*2^2+28*1^2 = 1*2^2+32*1^2 = 36*1^2, therefore a(6)=10.
%p A093115 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A093115      `if`(i<1, 0, b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i))))
%p A093115     end:
%p A093115 a:= proc(n) local r; r:= isqrt(n);
%p A093115       b(n^2, r-`if`(r^2>n, 1, 0))
%p A093115     end:
%p A093115 seq(a(n), n=0..50);  # _Alois P. Heinz_, Apr 15 2013
%t A093115 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n-i^2, i]]]]; a[n_] := (r = Sqrt[n] // Floor; b[n^2, r - If[r^2 > n, 1, 0]]); Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jul 29 2015, after _Alois P. Heinz_ *)
%Y A093115 Cf. A093116, A092362, A001156, A037444, A078134.
%Y A093115 Cf. A072925.
%Y A093115 Cf. A072213, A161407. [_Reinhard Zumkeller_, Jun 10 2009]
%K A093115 nonn
%O A093115 0,5
%A A093115 _Reinhard Zumkeller_, Mar 21 2004
%E A093115 More terms from _Vladeta Jovovic_, Mar 24 2004
%E A093115 Corrected a(0) by _Alois P. Heinz_, Apr 15 2013