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A304797
Expansion of x * (d/dx) Sum_{k>=0} k!*x^(k*(k+1)/2)/Product_{j=1..k} (1 - x^j).
3
0, 1, 2, 9, 12, 25, 66, 91, 152, 243, 570, 715, 1212, 1729, 2702, 5265, 6960, 10489, 15318, 22363, 31100, 57771, 72534, 109411, 151032, 219025, 293930, 421281, 680820, 883369, 1256010, 1727971, 2396000, 3235419, 4447506, 5894875, 9266580, 11691001, 16380470, 21774753
OFFSET
0,3
COMMENTS
Sum of all parts of all compositions (ordered partitions) of n into distinct parts.
FORMULA
a(n) = n*A032020(n).
MAPLE
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
`if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
end:
a:= n-> n*add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..50); # Alois P. Heinz, May 18 2018
MATHEMATICA
nmax = 39; CoefficientList[Series[x D[Sum[k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], x], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 18 2018
STATUS
approved