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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 (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
Sequence in context: A365732 A137356 A365734 * A183929 A365252 A003522
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 March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)