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#81 by Michael De Vlieger at Wed Nov 16 08:53:55 EST 2022
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#80 by Gus Wiseman at Wed Nov 16 03:25:55 EST 2022
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#79 by Gus Wiseman at Tue Nov 15 22:02:16 EST 2022
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Conjecture: Also the number of topologically series-reduced ordered rooted trees with n + 2 vertices. This would imply a(n) = A005043A284778(n) - A284778-1) + A005043(n). For example, the a(0) = 1 through a(5) = 14 trees are:
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approved
editing
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#78 by Michael De Vlieger at Tue Nov 15 17:52:00 EST 2022
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#77 by Robert C. Lyons at Tue Nov 15 17:07:46 EST 2022
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#76 by Robert C. Lyons at Tue Nov 15 17:07:27 EST 2022
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Also gives the number of simple permutations of each length that avoid the pattern 321 (i.e. ., are the union of two increasing sequences, and in one line notation contain no nontrivial block of values which form an interval). There are 2 such permutations of length 4, 2 of length 5, etc. - Michael Albert, Jun 20 2012
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proposed
editing
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#75 by Gus Wiseman at Tue Nov 15 16:50:30 EST 2022
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#74 by Gus Wiseman at Tue Nov 15 10:14:57 EST 2022
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From Gus Wiseman, Nov 15 2022: (Start)
Conjecture: Also the number of topologically series-reduced ordered rooted trees with n + 2 vertices. This would imply a(n) = A005043(n) - A284778(n). For example, the a(0) = 1 through a(5) = 14 trees are:
(o) . (ooo) (oooo) (ooooo) (oooooo)
((oo)) ((ooo)) ((oo)oo) ((oo)ooo)
((oooo)) ((ooo)oo)
(o(oo)o) ((ooooo))
(oo(oo)) (o(oo)oo)
(((oo)o)) (o(ooo)o)
((o(oo))) (oo(oo)o)
(oo(ooo))
(ooo(oo))
(((oo)oo))
(((ooo)o))
((o(oo)o))
((o(ooo)))
((oo(oo)))
(End)
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approved
editing
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#73 by R. J. Mathar at Tue Oct 12 07:50:41 EDT 2021
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#72 by R. J. Mathar at Tue Oct 12 07:50:26 EDT 2021
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Conjecture: -(D-finite with recurrence -(n+2)*a(n) +(n-1)*a(n-1) +(5*n-2)*a(n-2) +3*(n-1)a(n-3)=0. - R. J. Mathar, Nov 17 2011
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approved
editing
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