A339162 Number of compositions (ordered partitions) of n into distinct parts, the least being 1.

0, 1, 0, 2, 2, 2, 8, 8, 14, 14, 44, 44, 74, 98, 128, 272, 326, 470, 644, 932, 1106, 2234, 2552, 3800, 4958, 7070, 9068, 12140, 20042, 24674, 34256, 45632, 61814, 80630, 109316, 135572, 217778, 262298, 362744, 466664, 636494, 805454, 1085804, 1375388, 1776938, 2591762

Offset

0,4

Links

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

Index entries for sequences related to compositions

Formula

G.f.: Sum_{k>=1} k! * x^(k*(k + 1)/2) / Product_{j=1..k-1} (1 - x^j).

Example

a(6) = 8 because we have [5, 1], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 5], [1, 3, 2] and [1, 2, 3].

Maple

b:= proc(n, i, p) option remember;
      `if`(n=0, p!, `if`((i-1)*(i+2)/2<n, 0,
       add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
    end:
a:= n-> `if`(n<1, 0, b(n-1$2, 1)):
seq(a(n), n=0..55);  # Alois P. Heinz, Nov 25 2020

Mathematica

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

Crossrefs

Cf. A032020, A096765, A339163, A339164, A339165, A339166.

Sequence in context: A353245 A275928 A221072 * A360808 A168272 A222448

Adjacent sequences: A339159 A339160 A339161 * A339163 A339164 A339165

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 25 2020

Status

approved