A304797 Expansion of x * (d/dx) Sum_{k>=0} k!*x^(k*(k+1)/2)/Product_{j=1..k} (1 - x^j).

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

Author

Ilya Gutkovskiy, May 18 2018

Keywords

• nonn

Comments

Sum of all parts of all compositions (ordered partitions) of n into distinct parts.

Links

• Index entries for sequences related to compositions

Crossrefs

Cf. A001787, A032020, A066186, A066189, A097910, A303664.

Programs

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]

Formula

a(n) = n*A032020(n).