login
A339418
Number of compositions (ordered partitions) of n into an even number of squares.
3
1, 0, 1, 0, 1, 2, 1, 4, 2, 6, 9, 8, 20, 16, 35, 44, 55, 102, 105, 196, 242, 344, 540, 652, 1084, 1380, 2037, 2964, 3912, 6042, 7976, 11776, 16634, 22968, 33963, 46156, 67457, 94510, 133180, 192316, 266514, 385338, 540138, 767008, 1094576, 1534704, 2200821, 3094248
OFFSET
0,6
FORMULA
G.f.: 4 / (3 + 2 * theta_3(x) - theta_3(x)^2), where theta_3() is the Jacobi theta function.
a(n) = (A006456(n) + A317665(n)) / 2.
a(n) = Sum_{k=0..n} A006456(k) * A317665(n-k).
EXAMPLE
a(9) = 6 because we have [4, 1, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1], [1, 1, 4, 1, 1, 1], [1, 1, 1, 4, 1, 1], [1, 1, 1, 1, 4, 1] and [1, 1, 1, 1, 1, 4].
MAPLE
b:= proc(n, t) option remember; local r, f, g;
if n=0 then t else r, f, g:=$0..2; while f<=n
do r, f, g:= r+b(n-f, 1-t), f+2*g-1, g+1 od; r fi
end:
a:= n-> b(n, 1):
seq(a(n), n=0..50); # Alois P. Heinz, Dec 03 2020
MATHEMATICA
nmax = 47; CoefficientList[Series[4/(3 + 2 EllipticTheta[3, 0, x] - EllipticTheta[3, 0, x]^2), {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 03 2020
STATUS
approved