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A339165
Number of compositions (ordered partitions) of n into distinct parts, the least being 4.
8
0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 50, 56, 80, 110, 134, 164, 212, 242, 410, 464, 632, 806, 1118, 1292, 1724, 2042, 2594, 3752, 4448, 5726, 7382, 9524, 12020, 15122, 18602, 23264, 28424, 39830, 46670, 60476, 74780
OFFSET
0,10
FORMULA
G.f.: Sum_{k>=1} k! * x^(k*(k + 7)/2) / Product_{j=1..k-1} (1 - x^j).
EXAMPLE
a(15) = 8 because we have [11, 4], [6, 5, 4], [6, 4, 5], [5, 6, 4], [5, 4, 6], [4, 11], [4, 6, 5] and [4, 5, 6].
MAPLE
b:= proc(n, i, p) option remember;
`if`(n=0, p!, `if`((i-4)*(i+5)/2<n, 0,
add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
end:
a:= n-> `if`(n<4, 0, b(n-4$2, 1)):
seq(a(n), n=0..55); # Alois P. Heinz, Nov 25 2020
MATHEMATICA
nmax = 52; CoefficientList[Series[Sum[k! x^(k (k + 7)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Nov 25 2020
STATUS
approved