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A055303 Number of labeled rooted trees with n nodes and 2 leaves. 4

%I

%S 3,36,360,3600,37800,423360,5080320,65318400,898128000,13172544000,

%T 205491686400,3399953356800,59499183744000,1098446469120000,

%U 21341245685760000,435361411989504000,9305850181275648000

%N Number of labeled rooted trees with n nodes and 2 leaves.

%C 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

%C 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

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

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

%F 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

%p seq((n-1)!*(n-2)*(n-3)*(n-4)/144, n = 5..21); # _Zerinvary Lajos_, Apr 25 2008

%t 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 *)

%Y Column 2 of A055302.

%K nonn

%O 3,1

%A _Christian G. Bower_, May 11 2000

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Last modified December 6 21:48 EST 2019. Contains 329809 sequences. (Running on oeis4.)