A303664 Expansion of (1/(1 - x))*Sum_{k>=0} k!*x^(k*(k+1)/2)/Product_{j=1..k} (1 - x^j).

1, 2, 3, 6, 9, 14, 25, 38, 57, 84, 141, 206, 307, 440, 633, 984, 1419, 2036, 2887, 4064, 5619, 8370, 11667, 16424, 22717, 31478, 42783, 58386, 82701, 113162, 155029, 210770, 285645, 383688, 514497, 682922, 940327, 1256300, 1687365, 2245692, 2997183, 3955448, 5233315, 6854588, 8978175, 11998806

Offset

0,2

Comments

Partial sums of A032020.

Links

Table of n, a(n) for n=0..45.

Index entries for sequences related to compositions

Maple

T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
    end:
b:= n-> add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
a:= proc(n) option remember;
      `if`(n<0, 0, b(n)+a(n-1))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 28 2018

Mathematica

nmax = 45; CoefficientList[Series[1/(1 - x) Sum[k! x^(k (k + 1)/2)/Product[1 - x^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000079, A032020, A036469.

Sequence in context: A146937 A032231 A200273 * A190276 A113808 A337114

Adjacent sequences: A303661 A303662 A303663 * A303665 A303666 A303667

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 28 2018

Status

approved