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
KEYWORD
easy,nonn
AUTHOR
Vladeta Jovovic, Mar 18 2005
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jan 18 2019
STATUS
approved