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A265260
Number of partitions of n into even squares.
1
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
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
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