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A339417
Number of compositions (ordered partitions) of n into an odd number of triangular numbers.
3
0, 1, 0, 2, 0, 4, 1, 9, 3, 19, 12, 41, 33, 91, 92, 203, 238, 466, 602, 1080, 1493, 2536, 3661, 6001, 8902, 14278, 21554, 34094, 52013, 81602, 125297, 195582, 301475, 469193, 724881, 1126161, 1742206, 2703888, 4186276, 6493192, 10057553, 15594636, 24161364, 37455851
OFFSET
0,4
FORMULA
G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k*(k + 1)/2)) - 1 / Sum_{k>=0} x^(k*(k + 1)/2)).
a(n) = (A023361(n) - A106507(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023361(k) * A106507(n-k).
EXAMPLE
a(8) = 3 because we have [6, 1, 1], [1, 6, 1] and [1, 1, 6].
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+g, g+1 od; r fi
end:
a:= n-> b(n, 0):
seq(a(n), n=0..50); # Alois P. Heinz, Dec 03 2020
MATHEMATICA
nmax = 43; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}]) - 1/Sum[x^(k (k + 1)/2), {k, 0, nmax}]), {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 03 2020
STATUS
approved