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 A103198 Number of compositions of n into a square number of parts. 9
 1, 1, 1, 1, 2, 5, 11, 21, 36, 58, 94, 166, 331, 716, 1574, 3368, 6892, 13447, 25127, 45391, 80428, 142615, 259085, 491855, 982400, 2045001, 4352661, 9291361, 19609786, 40574017, 81973315, 161568281, 311062991, 586764281, 1089615033, 2005257849, 3688711427 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS From Gus Wiseman, Jan 17 2019: (Start) 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:   [6] .   [1 1] [1 1] [1 3] [3 1] [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]   [1 3] [3 1] [1 1] [1 1] [2 2] [1 2] [2 1] [1 2] [2 1] [1 1] (End) LINKS Alois P. Heinz, Table of n, a(n) for n = 0..3329 (terms n = 1..1000 from Vaclav Kotesovec) Vaclav Kotesovec, a(n+1)/a(n) as a graph FORMULA a(n) = Sum_{k>0} (x/(1-x))^(k^2). Binomial transform of the characteristic function of squares A010052, with 0th term omitted. - Carl Najafi, Sep 09 2011 a(n) = Sum_{k >= 0} binomial(n-1,k^2-1). - Gus Wiseman, Jan 17 2019 MAPLE b:= proc(n, t) option remember; `if`(n=0,       `if`(issqr(t), 1, 0), add(b(n-j, t+1), j=1..n))     end: a:= n-> b(n, 0): seq(a(n), n=0..40);  # Alois P. Heinz, Jan 18 2019 MATHEMATICA nmax = 40; Rest[CoefficientList[Series[-1/2 + EllipticTheta[3, 0, x/(1-x)]/2, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 03 2017 *) CROSSREFS Cf. A000290, A011782, A052467, A089299, A089333, A120732, A323433, A323519, A323525. Sequence in context: A113032 A100134 A137356 * A183929 A003522 A112805 Adjacent sequences:  A103195 A103196 A103197 * A103199 A103200 A103201 KEYWORD easy,nonn AUTHOR Vladeta Jovovic, Mar 18 2005 EXTENSIONS a(0)=1 prepended by Alois P. Heinz, Jan 18 2019 STATUS approved

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Last modified May 6 09:25 EDT 2021. Contains 343580 sequences. (Running on oeis4.)