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A055303 Number of labeled rooted trees with n nodes and 2 leaves. 4
3, 36, 360, 3600, 37800, 423360, 5080320, 65318400, 898128000, 13172544000, 205491686400, 3399953356800, 59499183744000, 1098446469120000, 21341245685760000, 435361411989504000, 9305850181275648000 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

a(n+1) is the sum of the zero moments over all permutations of n. E.g., a(4) is [1,2,3].[0,1,2] + [1,3,2].[0,1,2] + [2,1,3].[0,1,2] + [2,3,1].[0,1,2] + [3,1,2].[0,1,2] + [3,2,1].[0,1,2] = 8 + 7 + 7 + 5 + 5 + 4 = 36. - Jon Perry, Feb 20 2004

a(n) is the number of pairs of elements (p(i),p(j)) in an n-permutation such that i > j and p(i) < p(j) where j is not equal to i-1. Loosely speaking, we could say: the number of inversions that are not descents. A055303 + A001286 = A001809. For example, a(3)=3 from the permutations (given in one line notation): (2,3,1), (3,1,2), (3,2,1) we have the pairs (1,2), (2,3) and (1,3) respectively. - Geoffrey Critzer, Jan 06 2013

LINKS

Table of n, a(n) for n=3..19.

Index entries for sequences related to rooted trees

FORMULA

E.g.f.: x^3/(2*(1-x)^3).

a(n) = (n-2)!*t(n-2)*t(n-1) = (n-2)!*(n-2)*(n-1)^2*n/4 = n!*(n-2)*(n-1)/4 = n!*t(n-2)/2 - Jon Perry, Feb 22 2004

MAPLE

seq((n-1)!*(n-2)*(n-3)*(n-4)/144, n = 5..21); # Zerinvary Lajos, Apr 25 2008

MATHEMATICA

With[{nn=20}, Drop[CoefficientList[Series[x^3/(2(1-x)^3), {x, 0, nn}], x]Range[0, nn]!, 3]] (* Harvey P. Dale, Nov 22 2012 *)

CROSSREFS

Column 2 of A055302.

Sequence in context: A067444 A092648 A026121 * A274403 A068177 A249894

Adjacent sequences:  A055300 A055301 A055302 * A055304 A055305 A055306

KEYWORD

nonn

AUTHOR

Christian G. Bower, May 11 2000

STATUS

approved

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Last modified November 19 19:09 EST 2019. Contains 329323 sequences. (Running on oeis4.)