A318405 - OEIS

A318405

Rectangular array R read by antidiagonals: R(n,k) = F(n+1)^k - k*F(n-1)*F(n)^(k-1), where F(n) = A000045(n), the n-th Fibonacci number; n >= 0, k >= 1.

%I #20 Mar 16 2020 13:06:02

%S 0,1,1,1,1,1,1,1,2,2,1,1,5,5,3,1,1,12,15,13,5,1,1,27,49,71,34,8,1,1,

%T 58,163,409,287,89,13,1,1,121,537,2315,2596,1237,233,21,1,1,248,1739,

%U 12709,23393,18321,5205,610,34,1,1,503,5537,67919,205894,268893,124177,22105,1597,55

%N Rectangular array R read by antidiagonals: R(n,k) = F(n+1)^k - k*F(n-1)*F(n)^(k-1), where F(n) = A000045(n), the n-th Fibonacci number; n >= 0, k >= 1.

%C Row index n begins with 0, column index begins with 1.

%C R(n,k) is the number of Markov equivalence classes whose skeleton is a spider graph with k legs, each of which contains n nodes of degree at most two. See Corollary 4.2 in the paper by A. Radhakrishnan et al. below.

%H A. Radhakrishnan, L. Solus, and C. Uhler. <a href="https://arxiv.org/abs/1706.06091">Counting Markov equivalence classes for DAG models on trees</a>, arXiv:1706.06091 [math.CO], 2017; Discrete Applied Mathematics 244 (2018): 170-185.

%e The rectangular array R(n,k) begins:

%e n\k| 1 2 3 4 5 6 7

%e ---+-------------------------------------------------------------

%e 0 | 0 1 1 1 1 1 1

%e 1 | 1 1 1 1 1 1 1

%e 2 | 1 2 5 12 27 58 121

%e 3 | 2 5 15 49 163 537 1739

%e 4 | 3 13 71 409 2315 12709 67919

%e 5 | 5 34 287 2596 23393 205894 1769027

%e 6 | 8 89 1237 18321 268893 3843769 53573477

%e 7 | 13 233 5205 124177 2941661 67944057 1530787237

%o (Sage)

%o def R(n, k):

%o return fibonacci(n+1)^k-k*fibonacci(n-1)*fibonacci(n)^(k-1)

%Y Columns include A000045, A001519, A318376, A318404.

%Y Cf. A007984.

%K nonn,tabl,easy

%O 0,9

%A _Liam Solus_, Aug 26 2018