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A103198
Number of compositions of n into a square number of parts.
10
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
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 *)
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Mar 18 2005
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jan 18 2019
STATUS
approved