A055312
Number of labeled rooted trees with n nodes and 11 leaves.
Original entry on oeis.org
12, 319332, 571389000, 340433566200, 110475588787200, 24370164778901760, 4123399022616706560, 576081026260420262400, 69763488659968731264000, 7574128968889436413824000, 755406265592909947874457600
Offset: 12
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[Factorial(n)*(n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(99*n^9 - 8316*n^8 + 307494*n^7 - 6562248*n^6 + 88967571*n^5 - 793483724*n^4 + 4647393828*n^3 - 17199472048*n^2 + 36398452992*n - 33443020800)/14684322099363840000: n in [12..25]]; // Vincenzo Librandi, Jul 25 2014
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Table[n! * (n-11)*(n-10)*(n-9)*(n-8)*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(n-1)*(99*n^9 - 8316*n^8 + 307494*n^7 - 6562248*n^6 + 88967571*n^5 - 793483724*n^4 + 4647393828*n^3 - 17199472048*n^2 + 36398452992*n - 33443020800)/14684322099363840000,{n,12,25}] (* Vaclav Kotesovec, Jul 25 2014 *)
A217763
Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes with unicyclic components having exactly k nodes with degree 1; n>=3, 0<=k<=n-3.
Original entry on oeis.org
1, 3, 12, 12, 90, 120, 70, 600, 1800, 1200, 465, 4725, 19530, 31500, 12600, 3507, 42168, 211680, 529200, 529200, 141120, 30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440, 286884, 4460760, 30413880, 117573120, 266716800, 312439680, 152409600, 21772800
Offset: 3
....o-o..........o-o......
....| |..........|\ ......
....o-o..........o-o......
T(4,0)=3 because the graph on the left has 4 nodes and 0 nodes with degree 1. It has 3 labelings.
T(4,1)=12 because the graph on the right has 4 nodes and 1 node with degree 1. It has 12 labelings.
1,
3, 12,
12, 90, 120,
70, 600, 1800, 1200,
465, 4725, 19530, 31500, 12600,
3507, 42168, 211680, 529200, 529200, 141120,
30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440.
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nn=10;f[list_]:=Select[list,#>0&];t=Sum[Sum[n!/k! StirlingS2[n-1,n-k]y^k x^n/n!,{k,1,n}],{n,0,nn}];Map[Reverse,Map[f,Drop[Range[0,nn]!CoefficientList[Series[ Exp[Log[1/(1-t)]/2-t/2-t^2/4],{x,0,nn}],{x,y}],3]]]//Grid
A231536
Triangular array read by rows. T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} whose functional digraph has exactly k nodes such that no nonrecurrent element is mapped into it. n >= 1, 1 <= k <= n.
Original entry on oeis.org
1, 2, 2, 6, 15, 6, 24, 108, 100, 24, 120, 840, 1340, 705, 120, 720, 7200, 17400, 15150, 5466, 720, 5040, 68040, 231000, 296100, 171402, 46921, 5040, 40320, 705600, 3198720, 5644800, 4687536, 2015272, 444648, 40320, 362880, 7983360, 46569600, 108168480, 121144464, 73191888, 25011576, 4625361, 362880
Offset: 1
T(3,3) = 6 because we have: (1,2,3),(2,1,3),(3,2,1),(1,3,2),(2,3,1),(3,1,2). In these 6 functions represented as a word there are 3 (all) elements with zero nonrecurrent elements mapped to them.
1,
2, 2,
6, 15, 6,
24, 108, 100, 24,
120, 840, 1340, 705, 120,
720, 7200, 17400, 15150, 5466, 720
Column k=1 and main diagonal give:
A000142.
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nn=6;Map[Select[#,#>0&]&,Drop[Range[0,nn]!CoefficientList[Series[1/(1- (-x+x y-ProductLog[-Exp[x (-1+y)] x])),{x,0,nn}],{x,y}],1]]//Grid
A284417
Triangular array read by rows. T(n,k) is the number of rooted labeled trees on n nodes with exactly k vertices whose unique descendent is a leaf, n >= 1, 0 <= k <= floor((n-1)/2) + delta_{2,n}.
Original entry on oeis.org
1, 0, 2, 3, 6, 16, 48, 145, 420, 60, 1536, 4800, 1440, 19579, 65730, 31500, 840, 290816, 1053696, 698880, 53760, 4942305, 19332936, 16367400, 2388960, 15120, 94689280, 399052800, 410296320, 93542400, 2419200, 2020278931, 9146127870, 11044008360, 3526261200, 200415600, 332640, 47523053568, 230339788800, 319018106880, 133013422080, 12986265600, 127733760
Offset: 1
Triangle begins
1,
0, 2,
3, 6,
16, 48,
145, 420, 60,
1536, 4800, 1440,
19579, 65730, 31500, 840,
290816, 1053696, 698880, 53760,
...
T(3,1)=6 because there are 6 labeled rooted trees (paths) o-o-o and these 6 trees have 1 vertex whose only descendent is a leaf. T(3,0) = 3 because there are 3 labeled trees of the form
o
/ \
o o
and these 3 trees have no such vertices.
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nn = 10; Range[0, nn]! CoefficientList[Series[-z^2 + u z^2 - ProductLog[-E^((-1 + u) z^2) z], {z, 0, nn}], {z, u}] // Grid
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