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

%I A097910 #34 Aug 10 2020 08:59:19
%S A097910 1,1,5,5,9,27,31,49,71,185,207,339,457,685,1421,1745,2577,3615,5143,
%T A097910 6877,13439,15965,23823,31983,45553,59425,83549,139013,173769,244803,
%U A097910 330391,452257,597935,810929,1052559,1692723,2074321,2890333,3783821,5178041,6658377
%N A097910 Number of parts in all compositions of n into distinct parts.
%H A097910 Alois P. Heinz, <a href="/A097910/b097910.txt">Table of n, a(n) for n = 1..5000</a>
%F A097910 G.f.: Sum(k >= 0; k*k! x^((k^2+k)/2) / Prod(1<=j<=k; 1-x^j)).
%F A097910 a(n) = Sum_{k=1..floor((sqrt(8*n+1)-1)/2)} k! * k * A008289(n,k). - _Alois P. Heinz_, Aug 10 2020
%p A097910 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A097910       `if`(n>i*(i+1)/2, [][], zip((x, y)->x+y, [b(n, i-1)],
%p A097910       `if`(i>n, [], [0, b(n-i, i-1)]), 0)[]))
%p A097910     end:
%p A097910 a:= n-> (l-> add(i*l[i+1]*i!, i=1..nops(l)-1))([b(n$2)]):
%p A097910 seq(a(n), n=1..50);  # _Alois P. Heinz_, Nov 20 2012
%p A097910 # second Maple program:
%p A097910 b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0,
%p A097910      `if`(n=0, p!*p, b(n-i, min(n-i, i-1), p+1)+b(n, i-1, p)))
%p A097910     end:
%p A097910 a:= n-> b(n$2, 0):
%p A097910 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 10 2020
%t A097910 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 *)
%Y A097910 Cf. A001792, A008289, A015723, A032020, A072574, A336875.
%K A097910 easy,nonn
%O A097910 1,3
%A A097910 _Vladeta Jovovic_, Sep 04 2004
%E A097910 More terms from _Robert G. Wilson v_ and _John W. Layman_, Sep 08 2004