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A089333 Number of partitions into a square number of parts.

%I A089333 #16 Jan 10 2016 10:03:12
%S A089333 1,1,1,1,2,2,3,4,6,8,11,14,19,24,31,39,51,63,80,99,124,153,190,233,
%T A089333 288,353,432,527,643,780,947,1145,1383,1665,2002,2399,2874,3431,4090,
%U A089333 4865,5779,6847,8103,9568,11283,13280,15610,18313,21462,25108,29337,34227
%N A089333 Number of partitions into a square number of parts.
%C A089333 Also number of partitions of n such that the largest part is a square. Example: a(7)=4 because we have [4,3], [4,2,1], [4,1,1,1] and [1,1,1,1,1,1,1]. - _Emeric Deutsch_, Apr 04 2006
%H A089333 Alois P. Heinz, <a href="/A089333/b089333.txt">Table of n, a(n) for n = 0..10000</a>
%F A089333 G.f.: Sum(x^(n^2)/Product(1-x^i, i = 1 .. n^2), n = 1 .. infinity).
%e A089333 a(7)=4 because we have [7], [4,1,1,1], [3,2,1,1] and [2,2,2,1].
%p A089333 g:=sum(x^(k^2)/product(1-x^i,i=1..k^2),k=1..7): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..51); # _Emeric Deutsch_, Apr 04 2006
%p A089333 # second Maple program:
%p A089333 b:= proc(n, i) option remember; `if`(n<0, 0,
%p A089333       `if`(n=0 or i=1, 1, `if`(i<1, 0, b(n, i-1)+
%p A089333       `if`(i>n, 0, b(n-i, i)))))
%p A089333     end:
%p A089333 a:= n-> add(b(n-i^2, i^2), i=0..isqrt(n)):
%p A089333 seq(a(n), n=0..60);  # _Alois P. Heinz_, Sep 24 2015
%t A089333 b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0 || i == 1, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]]; a[n_] := Sum[b[n - i^2, i^2], {i, 0, Sqrt[n]}]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Jan 10 2016, after _Alois P. Heinz_*)
%K A089333 easy,nonn
%O A089333 0,5
%A A089333 _Vladeta Jovovic_, Dec 25 2003
%E A089333 a(0)=1 from _Alois P. Heinz_, Sep 24 2015