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

Author

Emeric Deutsch, Jan 26 2016

Keywords

• nonn

Comments

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

Examples

			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].
		

Links

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

Crossrefs

Cf. A001156, A042968, A167661.

Programs

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 *)

Formula

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

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

Extensions

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