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A097910 Number of parts in all compositions of n into distinct parts. 7
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 30 07:58 EDT 2021. Contains 346348 sequences. (Running on oeis4.)