cp's OEIS Frontend

A083410 a(n) = A083385(n)/n.

%I A083410 #31 Feb 20 2025 11:20:00
%S A083410 1,4,22,154,1306,12994,148282,1908274,27333706,431220034,7428550042,
%T A083410 138737478994,2792050329706,60231133487074,1386484468239802,
%U A083410 33921605427779314,878976357571495306,24046780495646314114,692622345890928153562,20950628198687114521234,663992311200423614606506
%N A083410 a(n) = A083385(n)/n.
%C A083410 From _Michael Somos_, Mar 04 2004: (Start)
%C A083410 Stirling transform of A052849(n+1)=[4,12,48,240,...] is 4*a(n)=[4,16,88,616,...].
%C A083410 Stirling transform of A001710(n+1)=[1,3,12,160,...] is a(n)=[1,4,22,154,...].
%C A083410 Stirling transform of A001563(n+1)=[4,18,96,600,...] is a(n+1)=[4,22,154,...]. (End)
%H A083410 N. J. A. Sloane and Thomas Wieder, <a href="https://arxiv.org/abs/math/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
%F A083410 E.g.f.: (1/(2-exp(x))^2-1)/2. - _Michael Somos_, Mar 04 2004
%F A083410 G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k+4) - 2*x^2*(k+1)*(k+3)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 03 2013
%F A083410 a(n) ~ n! * n / (8 * (log(2))^(n+2)). - _Vaclav Kotesovec_, Jul 01 2018
%F A083410 a(n) = Sum_{k=1..n} k * A090665(n,k). - _Alois P. Heinz_, Feb 20 2025
%p A083410 b:= proc(n, m) option remember;
%p A083410      `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
%p A083410     end:
%p A083410 a:= n-> b(n, 0)/2:
%p A083410 seq(a(n), n=1..23);  # _Alois P. Heinz_, Feb 14 2025
%t A083410 a[n_] := (-1)^n (PolyLog[-n - 1, 2] - PolyLog[-n, 2])/8;
%t A083410 Array[a, 21] (* _Jean-François Alcover_, Sep 10 2018, from A005649 *)
%o A083410 (PARI) a(n)=if(n<0,0,n!*polcoeff(subst((1/(1-y)^2-1)/2,y,exp(x+x*O(x^n))-1),n))
%Y A083410 A005649(n)=2*a(n), if n>0.
%Y A083410 Pairwise sums of A091346.
%Y A083410 Cf. A090665.
%K A083410 nonn
%O A083410 1,2
%A A083410 _N. J. A. Sloane_, Jun 08 2003