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