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A055303
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Number of labeled rooted trees with n nodes and 2 leaves.
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5
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3, 36, 360, 3600, 37800, 423360, 5080320, 65318400, 898128000, 13172544000, 205491686400, 3399953356800, 59499183744000, 1098446469120000, 21341245685760000, 435361411989504000, 9305850181275648000, 208013121699102720000, 4853639506312396800000
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OFFSET
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3,1
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COMMENTS
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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
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LINKS
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FORMULA
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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 where t = A000217. - Jon Perry, Feb 22 2004
D-finite with recurrence: (n-3)*a(n) - (n^2 - n)*a(n-1) = 0. - Georg Fischer, Aug 17 2021
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MAPLE
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f:= gfun:-rectoproc({(n-3)*a(n) - (n^2-n)*a(n-1), a(3)=3}, a(n), remember): map(f, [$3..20]); # Georg Fischer, Aug 17 2021
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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