A090665 -id:A090665 - cp's OEIS frontend

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

1, 4, 22, 154, 1306, 12994, 148282, 1908274, 27333706, 431220034, 7428550042, 138737478994, 2792050329706, 60231133487074, 1386484468239802, 33921605427779314, 878976357571495306, 24046780495646314114, 692622345890928153562, 20950628198687114521234, 663992311200423614606506

Offset: 1

N. J. A. Sloane, Jun 08 2003

nonn

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of A052849(n+1)=[4,12,48,240,...] is 4*a(n)=[4,16,88,616,...].
Stirling transform of A001710(n+1)=[1,3,12,160,...] is a(n)=[1,4,22,154,...].
Stirling transform of A001563(n+1)=[4,18,96,600,...] is a(n+1)=[4,22,154,...]. (End)

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

A005649(n)=2*a(n), if n>0.
Pairwise sums of A091346.
Cf. A090665.

b:= proc(n, m) option remember;
     `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0)/2:
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 14 2025

a[n_] := (-1)^n (PolyLog[-n - 1, 2] - PolyLog[-n, 2])/8;
Array[a, 21] (* Jean-François Alcover, Sep 10 2018, from A005649 *)

a(n)=if(n<0,0,n!*polcoeff(subst((1/(1-y)^2-1)/2,y,exp(x+x*O(x^n))-1),n))

E.g.f.: (1/(2-exp(x))^2-1)/2. - Michael Somos, Mar 04 2004
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
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n) = Sum_{k=1..n} k * A090665(n,k). - Alois P. Heinz, Feb 20 2025

A054255 Triangle T(n,k) (n >= 1, 0<=k<=n) giving number of preferential arrangements of n things beginning with k (transposed, then read by rows).

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 6, 18, 25, 26, 24, 84, 134, 149, 150, 120, 480, 870, 1050, 1081, 1082, 720, 3240, 6600, 8700, 9302, 9365, 9366, 5040, 25200, 57120, 82320, 92526, 94458, 94585, 94586, 40320, 221760, 554400, 871920, 1038744, 1085364, 1091414, 1091669, 1091670

Offset: 1

Eugene McDonnell (Eemcd(AT)aol.com), May 05 2000

Can be generated from Stirling_2 triangle A008277 (cf. A028246, which is intermediate between the two arrays).

			   1;
   1,  2;
   2,  5,   6;
   6, 18,  25,  26;
  24, 84, 134, 149, 150;
  ...

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Row sums give A000670. First 3 rows are A000629, A002050 = A000629 - 1, 2*A002051 = (A000629 - 2^m) (m >= 0).

Cf. A090665 (triangle with rows reversed).

More terms from James Sellers, May 05 2000

cp's OEIS Frontend

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

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