A097910 Number of parts in all compositions of n into distinct parts.

1, 1, 5, 5, 9, 27, 31, 49, 71, 185, 207, 339, 457, 685, 1421, 1745, 2577, 3615, 5143, 6877, 13439, 15965, 23823, 31983, 45553, 59425, 83549, 139013, 173769, 244803, 330391, 452257, 597935, 810929, 1052559, 1692723, 2074321, 2890333, 3783821, 5178041, 6658377

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Formula

G.f.: Sum(k >= 0; k*k! x^((k^2+k)/2) / Prod(1<=j<=k; 1-x^j)).

a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} k! * k * A008289(n,k). - Alois P. Heinz, Aug 10 2020

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(n>i*(i+1)/2, [][], zip((x, y)->x+y, [b(n, i-1)],
      `if`(i>n, [], [0, b(n-i, i-1)]), 0)[]))
    end:
a:= n-> (l-> add(i*l[i+1]*i!, i=1..nops(l)-1))([b(n$2)]):
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 20 2012
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
     `if`(n=0, p!*p, b(n-i, min(n-i, i-1), p+1)+b(n, i-1, p)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 10 2020

Mathematica

Drop[ CoefficientList[ Series[ Sum[ k*k!*x^((k^2 + k)/2)/Product[1 - x^j, {j, 1, k}], {k, 1, 45}], {x, 0, 40}], x], 1] (* Robert G. Wilson v, Sep 08 2004 *)

Crossrefs

Cf. A001792, A008289, A015723, A032020, A072574, A336875.

Sequence in context: A323301 A147047 A173322 * A321655 A336128 A049122

Adjacent sequences: A097907 A097908 A097909 * A097911 A097912 A097913

Keyword

easy,nonn

Author

Vladeta Jovovic, Sep 04 2004

Extensions

More terms from Robert G. Wilson v and John W. Layman, Sep 08 2004

Status

approved