A265260 Number of partitions of n into even squares.

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 5, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 6, 0, 0, 0, 8, 0, 0, 0, 9, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 12, 0, 0, 0, 13, 0, 0, 0, 14, 0, 0, 0, 14, 0, 0, 0, 16, 0, 0, 0, 19, 0, 0, 0, 20, 0

Offset

0,17

Comments

a(n) = 0 if and only if n is not divisible by 4 (sequence A042968).

Links

Hans Havermann, Table of n, a(n) for n = 0..831

Formula

G.f.: 1/Product_{i>=1} (1 - x^{4i^2}).

a(4n) = A001156(n). - Alois P. Heinz, Jan 27 2016

Example

a(28) = 2 because we have [4,4,4,4,4,4,4] and [4,4,4,16].
a(32) = 3 because we have [4,4,4,4,4,4,4,4], [4,4,4,4,16], and [16,16].

Maple

g := 1/mul(1-x^(4*i^2), i = 1 .. 150): gser := series(g, x = 0, 105): seq(coeff(gser, x, n), n = 0 .. 100);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i))))
    end:
a:= n-> `if`(irem(n, 4, 'm')=0, b(m, isqrt(m)), 0):
seq(a(n), n=0..120);  # Alois P. Heinz, Jan 27 2016

Mathematica

a[n_] := If[n==0, 1, If[Divisible[n, 4], PowersRepresentations[n/4, n/4, 2] // Length, 0]]; Array[a, 100, 0] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)

Crossrefs

Cf. A001156, A042968, A167661.

Sequence in context: A083891 A363860 A087623 * A363888 A338268 A269243

Adjacent sequences: A265257 A265258 A265259 * A265261 A265262 A265263

Keyword

nonn

Author

Emeric Deutsch, Jan 26 2016

Extensions

Data-section extended up to a(105) by Antti Karttunen, Nov 21 2017, from the b-file provided by Hans Havermann

Status

approved