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A201199 Triangle version of the array w(N,L) of the total number of round trips of length L on closed Laguerre graphs Lc_N.

%I A201199 #12 Aug 27 2019 23:34:02
%S A201199 1,1,2,1,4,3,1,18,9,4,1,76,53,16,5,1,322,357,120,25,6,1,1364,2489,
%T A201199 1024,233,36,7,1,5778,17509,9424,2545,404,49,8,1,24476,123449,89536,
%U A201199 29985,5400,645,64,9,1,103682,870893,862560,367505,78392,10213,968,81,10
%N A201199 Triangle version of the array w(N,L) of the total number of round trips of length L on closed Laguerre graphs Lc_N.
%C A201199 For Laguerre graphs (open and closed ones) see the W. Lang link on Jacobi graphs under A201198. There one also finds a sketch of the closed Laguerre graph Lc_4 as Fig.4.
%C A201199 The total number of round trips on the closed Laguerre graph Lc_N, for N>=3, with N vertices N^2 loops, binomial(N,2) lines between neighboring vertices and two lines between the first and the last vertex (in total (3*N-1)*N/2+2 = (3*N^2-N+4)/2  lines) is w(N,L) = sum(w(N,L;p_n->p_n),n=1..N) =  Trace((L_N)^L) = sum((x_n^{(N)})^L,n=1..N), with the N x N  symmetric adjacency matrix, also called  Lc_N, having non-vanishing elements (Lc_N)[n,n] = 2*n-1, n=1..N, (Lc_N)[n,n+1] = (Lc_N)[n+1,n] = n, n=1..N-1, and (Lc_N)[1,N]= 2=(Lc_N)[N,1].  The eigenvalues of Lc_N are x_n^{(N)}. They are the  zeros of the characteristic polynomial Lac_N(x):=Det(x*1_N -Lc_N) with the N x N unit matrix 1_N. These are the polynomials Lac_N(x) = La(N,x) - 4*La1(N-2,x) - 4*(N-1)!, with the ordinary monic Laguerre polynomials La(N,x) with coefficient array given by A021009(n,m)*(-1)^n  and the first associated monic Laguerre polynomials La1(N-2,x) with coefficient array given by A199577(n,m).  For N=1 one has  Lc_1=L_1 (Laguerre graph with one vertex and one loop) with L_1(x)=x-1, and for N=2 one has a graph where one vertex has one loop, the other three, and there are two lines joining these vertices, hence Lc_2(x)= x^2-4*x-1.
%H A201199 Wolfdieter Lang, <a href="/A201198/a201198_1.pdf">Counting walks on Jacobi graphs: an application of orthogonal polynomials.</a>
%F A201199 a(K,N) =  w(N,K-N+1),K>=0, N=1,...,K+1, with w(N,L) the total number of round trips of length L on the closed Laguerre graph Lc_N described above in the comment section.
%F A201199 The o.g.f. of w(N,L) is: G(N,x)=y*(d/dx)Lac_N(x)/Lac_N(x) with y=1/x.
%F A201199   The characteristic polynomial Lac_N(x) has also been given in the comment section above.
%e A201199 The array w(N,L) starts:
%e A201199 N\L 0   1    2     3      4        5         6  ...
%e A201199 1:  1   1    1     1      1        1         1
%e A201199 2:  2   4   12    40    136      464      1584
%e A201199 3:  3   9   53   357   2489    17509    123449
%e A201199 4:  4  16  120  1024   9424    89536    862560
%e A201199 5:  5  25  233  2545  29985   367505   4599521
%e A201199 6:  6  36  404  5400  78392  1188336  18460016
%e A201199 7:  7  49  645 10213 176473  3195829  59473593
%e A201199 8:  8  64  968 17728 355536  7493504 162671840
%e A201199 9:  9  81 1385 28809 657953 15826041 392792273
%e A201199 ...The triangle a(K,N) = w(N,K-N+1) starts:
%e A201199 K\N 1      2       3      4      5     6     7   8  9..
%e A201199 0:  1
%e A201199 1:  1      2
%e A201199 2:  1      4       3
%e A201199 3:  1     18       9      4
%e A201199 4:  1     76      53     16      5
%e A201199 5:  1    322     357    120     25     6
%e A201199 6:  1   1364    2489   1024    233    36     7
%e A201199 7:  1   5778   17509   9424   2545   404    49   8
%e A201199 8:  1  24476  123449  89536  29985  5400   645  64  9
%e A201199 ...
%e A201199 For the graph Lc_4, shown in the W. Lang link as Figure 4, the counting for round trips of length L=2 for each of the four vertices V_i, i=1..4, read from left to right, is as follows.
%e A201199 V_1: 1+1+(1+1+2*1), V_2: 3+2*binomial(3,2)+1+(1+1+2*1),
%e A201199 V_3: 5+2*binomial(5,2)+(1+1+2*1)+(3+2*binomial(3,2)),
%e A201199 V_4: 7+2*binomial(7,2)+(3+2*binomial(3,2))+(1+1+2*1),
%e A201199 this sums to the total number  w(4,2)= 120  =  a(5,4).
%e A201199 Compared to the open L_4 graph (see the corresponding A201198 entry 4*28 = 112) one has to add 2*(1+1+2*1)=8 from the new two lines joining V_1 and V_4.
%Y A201199 Cf. A201198 (open Laguerre graphs).
%K A201199 nonn,easy,walk,tabl
%O A201199 0,3
%A A201199 _Wolfdieter Lang_, Nov 30 2011