A103198 Number of compositions of n into a square number of parts.

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 *)

Crossrefs

Cf. A000290, A011782, A052467, A089299, A089333, A120732, A323433, A323519, A323525.

Sequence in context: A365732 A137356 A365734 * A183929 A365252 A003522

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