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A103198 Number of compositions of n into a square number of parts.

%I A103198 #27 Dec 17 2022 21:31:09
%S A103198 1,1,1,1,2,5,11,21,36,58,94,166,331,716,1574,3368,6892,13447,25127,
%T A103198 45391,80428,142615,259085,491855,982400,2045001,4352661,9291361,
%U A103198 19609786,40574017,81973315,161568281,311062991,586764281,1089615033,2005257849,3688711427
%N A103198 Number of compositions of n into a square number of parts.
%C A103198 From _Gus Wiseman_, Jan 17 2019: (Start)
%C A103198 Also the number of ways to fill a square matrix with the parts of an integer partition of n. For example, the a(6) = 11 matrices are:
%C A103198   [6]
%C A103198 .
%C A103198   [1 1] [1 1] [1 3] [3 1] [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
%C A103198   [1 3] [3 1] [1 1] [1 1] [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
%C A103198 (End)
%H A103198 Alois P. Heinz, <a href="/A103198/b103198.txt">Table of n, a(n) for n = 0..3329</a> (terms n = 1..1000 from Vaclav Kotesovec)
%H A103198 Vaclav Kotesovec, <a href="/A103198/a103198.jpg">a(n+1)/a(n) as a graph</a>
%F A103198 a(n) = Sum_{k>=0} (x/(1-x))^(k^2).
%F A103198 Binomial transform of the characteristic function of squares A010052, with 0th term omitted. - _Carl Najafi_, Sep 09 2011
%F A103198 a(n) = Sum_{k >= 0} binomial(n-1,k^2-1). - _Gus Wiseman_, Jan 17 2019
%p A103198 b:= proc(n, t) option remember; `if`(n=0,
%p A103198       `if`(issqr(t), 1, 0), add(b(n-j, t+1), j=1..n))
%p A103198     end:
%p A103198 a:= n-> b(n, 0):
%p A103198 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jan 18 2019
%t A103198 nmax = 40; Rest[CoefficientList[Series[-1/2 + EllipticTheta[3, 0, x/(1-x)]/2, {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jan 03 2017 *)
%Y A103198 Cf. A000290, A011782, A052467, A089299, A089333, A120732, A323433, A323519, A323525.
%K A103198 easy,nonn
%O A103198 0,5
%A A103198 _Vladeta Jovovic_, Mar 18 2005
%E A103198 a(0)=1 prepended by _Alois P. Heinz_, Jan 18 2019