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A103198 Number of compositions of n into a square number of parts. 3
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,changed

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 January 19 16:21 EST 2019. Contains 319307 sequences. (Running on oeis4.)