This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A055303 #29 Aug 17 2021 19:20:19
%S A055303 3,36,360,3600,37800,423360,5080320,65318400,898128000,13172544000,
%T A055303 205491686400,3399953356800,59499183744000,1098446469120000,
%U A055303 21341245685760000,435361411989504000,9305850181275648000,208013121699102720000,4853639506312396800000
%N A055303 Number of labeled rooted trees with n nodes and 2 leaves.
%C A055303 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 A055303 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 A055303 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A055303 E.g.f.: x^3/(2*(1-x)^3).
%F A055303 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
%F A055303 D-finite with recurrence: (n-3)*a(n) - (n^2 - n)*a(n-1) = 0. - _Georg Fischer_, Aug 17 2021
%F A055303 a(n) = 3 * A001754(n). - _Alois P. Heinz_, Aug 17 2021
%p A055303 seq(n!*(n-2)*(n-1)/4, n = 3..21); # _Zerinvary Lajos_, Apr 25 2008 [corrected by _Georg Fischer_, Aug 17 2021]
%p A055303 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
%t A055303 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 A055303 Column 2 of A055302.
%Y A055303 Cf. A000217, A001754.
%K A055303 nonn,easy
%O A055303 3,1
%A A055303 _Christian G. Bower_, May 11 2000