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Number of partitions of n^2 into squares not greater than n.
5

%I #14 Dec 29 2016 16:08:55

%S 1,1,1,1,5,7,10,13,17,108,159,228,317,430,572,748,5753,8125,11266,

%T 15376,20672,27430,35942,46575,59717,523905,708028,946875,1253880,

%U 1645224,2140099,2761318,3535658,4494602,5674753,7118724,69766770,90940578,117756370

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

%H Alois P. Heinz, <a href="/A093115/b093115.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%p end:

%p a:= proc(n) local r; r:= isqrt(n);

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

%p end:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Apr 15 2013

%t 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 Cf. A093116, A092362, A001156, A037444, A078134.

%Y Cf. A072925.

%Y Cf. A072213, A161407. [_Reinhard Zumkeller_, Jun 10 2009]

%K nonn

%O 0,5

%A _Reinhard Zumkeller_, Mar 21 2004

%E More terms from _Vladeta Jovovic_, Mar 24 2004

%E Corrected a(0) by _Alois P. Heinz_, Apr 15 2013