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 A201198 Triangle version of the array w(N,L) of the average number of round trips of length L on Laguerre graphs L_N. 6
 1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 15, 4, 1, 1, 68, 87, 28, 5, 1, 1, 232, 531, 232, 45, 6, 1, 1, 792, 3303, 2056, 485, 66, 7, 1, 1, 2704, 20691, 18784, 5645, 876, 91, 8, 1, 1, 9232, 129951, 174112, 68245, 12636, 1435, 120, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS For Laguerre graphs see the W. Lang link on Jacobi graphs (named after the symmetric tridiagonal Jacobi adjacency matrices, related to orthogonal polynomials). There one also finds a sketch of the Laguerre graph L_4 in Fig. 3. The average number of round trips for the Laguerre graph L_N  with N vertices, N^2 loops and binomial(N,2) lines between neighboring vertices (in total (3*N-1)*N/2 lines) is w(N,L) = sum(w(N,L;p_n->p_n), n=1..N)/N =  Trace((L_N)^L)/N = sum((x_n^{(N)})^L,n = 1..N)/N, with the N x N tridiagonal symmetric adjacency matrix L_N, having non-vanishing elements (L_N)[n,n] = 2*n-1, n=1..N, (L_N)[n,n+1] = (L_N)[n+1,n] = n, n=1..N-1. The eigenvalues of L_N are x_n^{(N)}. They are the  zeros of the characteristic polynomial La_N(x):=Det(x*1_N -L_N) with the N x N unit matrix 1_N. These are the ordinary monic Laguerre polynomials with coefficient triangle given in A021009(n,m)*(-1)^n. LINKS Eric W. Weisstein, from MathWorld: Laguerre Polynomial. FORMULA a(K,N) = w(N,K-N+1), with w(N,L) the total number of round trips of length L on the Laguerre graph L_N divided by N (average length L round trip numbers). The definition of the graph L_N is given as a comment above. The o.g.f. of w(N,L) is G(N,x) = (1/N)*y*(d/dx)La_N(x)/La_N(x)) with y=1/x. This can be written as   G(N,x)= 1 + N*La_{N-1}(1/x)/La_N(1/x), where La_N(x) are the monic Laguerre polynomials (see a comment above). EXAMPLE The array w(N,L) starts: N\L 0 1   2          4       5        6          7 ... 1:  1 1   1    1     1       1        1          1 2:  1 2   6   20    68     232      792       2704 3:  1 3  15   87   531    3303    20691     129951 4:  1 4  28  232  2056   18784   174112    1625152 5:  1 5  45  485  5645   68245   841725   10495525 6:  1 6  66  876 12636  190296  2935656   45927216 7:  1 7  91 1435 24703  445627  8259727  155635459 8:  1 8 120 2192 43856  922048 19964736  440311936 9:  1 9 153 3177 72441 1739529 43098777 1089331497 ... The triangle a(K,N) = w(N,K-N+1) starts: K\N 1    2      3      4     5     6    7   8 9 10 ... 0:  1 1:  1    1 2:  1    2      1 3:  1    6      3      1 4:  1   20     15      4     1 5:  1   68     87     28     5     1 6:  1  232    531    232    45     6    1 7:  1  792   3303   2056   485    66    7   1 8:  1 2704  20691  18784  5645   876   91   8 1 9:  1 9232 129951 174112 68245 12636 1435 120 9  1 ... For the graph L_4, shown in the W. Lang link as Figure 3, the counting for round trips of length L=2 for each of the four vertices V_i, i=1..4, reads, from left to right, as follows. V_1: 1+1, V_2: 3+2*binomial(3,2)+1+(1+1+2*1), V_3: 5+2*binomial(5,2)+(1+1+2*1)+(3+2*binomial(3,2)), V_4: 7+2*binomial(7,2)+(3+2*binomial(3,2)), this sums to 112, hence the average number is w(4,2)= 112/4 = 28 = a(5,4). CROSSREFS A201199 (closed Laguerre graphs). Sequence in context: A332405 A332403 A263341 * A120258 A201922 A181644 Adjacent sequences:  A201195 A201196 A201197 * A201199 A201200 A201201 KEYWORD nonn,easy,walk,tabl AUTHOR Wolfdieter Lang, Nov 30 2011 STATUS approved

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Last modified June 21 17:44 EDT 2021. Contains 345365 sequences. (Running on oeis4.)