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A339166
Number of compositions (ordered partitions) of n into distinct parts, the least being 5.
8
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 26, 56, 56, 86, 110, 140, 164, 218, 242, 296, 464, 518, 686, 884, 1172, 1370, 1802, 2120, 2672, 3134, 4526, 5108, 6764, 8186, 10682, 13088, 16544, 19790, 24950, 29876, 36716
OFFSET
0,12
FORMULA
G.f.: Sum_{k>=1} k! * x^(k*(k + 9)/2) / Product_{j=1..k-1} (1 - x^j).
EXAMPLE
a(18) = 8 because we have [13, 5], [7, 6, 5], [7, 5, 6], [6, 7, 5], [6, 5, 7], [5, 13], [5, 7, 6] and [5, 6, 7].
MAPLE
b:= proc(n, i, p) option remember;
`if`(n=0, p!, `if`((i-5)*(i+6)/2<n, 0,
add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
end:
a:= n-> `if`(n<5, 0, b(n-5$2, 1)):
seq(a(n), n=0..60); # Alois P. Heinz, Nov 25 2020
MATHEMATICA
nmax = 55; CoefficientList[Series[Sum[k! x^(k (k + 9)/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