cp's OEIS Frontend

Hankel transform is 2^n. Successive binomial transforms are A002426, A000984, A026375, A081671, A098409, A098410.

From Andrew V. Sutherland, Feb 29 2008: (Start)

Counts returning walks of length n on a 1-d integer lattice with step set {-1,+1}.

Moment sequence of the trace of a random matrix in G = SO(2). If X = tr(A) is a random variable (A distributed with Haar measure on G), then a(n) = E[X^n].

Also the moment sequence of the trace of the k-th power of a random matrix in USp(2) = SU(2), for all k > 2.

(End)

From Paul Barry, Aug 10 2009: (Start)

The Hankel transform of 0,1,0,2,0,6,... is 0,-1,0,4,0,-16,0,... with general term I*(-4)^(n/2)(1 - (-1)^n)/4, I = sqrt(-1).

The Hankel transform of 1,1,0,2,0,6,... (which has g.f. 1 + x/sqrt(1 - 4*x^2)) is A164111. (End)

a(n) = A204293(2*n,n): central terms of the triangle in A204293. - Reinhard Zumkeller, Jan 14 2012

a(n) is the total number of closed walks (round trips) of length n on the graph P_N (a line with N nodes and N-1 edges), divided by N, in the limit N -> infinity. See a comment on A198632 and a link under A201198. - Wolfdieter Lang, Oct 10 2012

This array is an example of counting walks on a graph whose adjacency matrix is given by a special symmetric tridiagonal matrix with nonnegative integer entries, appropriate for orthogonal polynomials. These are quadratic Jacobi matrices J_n with nonnegative entries. The corresponding graphs could be called Jacobi graphs. Here Chebyshev S-polynomials (coefficients A049310) are considered, which belong to the Jacobi class of the classical orthogonal polynomial systems. The corresponding graph has adjacency matrix [[0,1,0,...],[1,0,1,...],[0,1,0,1,...]...[0,...0,1,0]] (n rows and n columns), with characteristic polynomial S(n,x) (see also a comment by Michael Somos on A049310).

w(n,l;p_k->p_m) = ((J_n)^l)(k,m) is the number of walks of length l from vertex p_k to vertex p_m on such a Jacobi graph. w(n,0; p_k->p_m) = delta(k,m), with the Kronecker symbol delta. The total number of closed walks of length l is w(n,l):=Sum_{i=1..n} w(n,l; p_i->p_i) = trace(J_n^l), which is the l-th power sum of the eigenvalues of J_n, i.e., the zeros of the characteristic polynomial for J_n. There are theorems for the o.g.f. of the normalized power sums of these zeros. See, e.g., the given W. Lang reference, p. 244. This results for the S-polynomial in the o.g.f. G(n,x) = Sum_{l=0..infinity} w(n,l)*x^l = y*(d/dy)S(n,y)/S(y) with y=1/x. This can be rewritten in the form given in the formula section (this results from eq. (3.8b) of the W. Lang reference, and in eq. (3.8d) it should be coth, not tanh).

From Wolfdieter Lang, Oct 10 2012: (Start)

For an accompanying paper on path counting on Jacobi graphs see the W. Lang link under A201198.

The total number of round trips of length L on the graph P_n, taken per site, becomes for n -> infinity A126869(L). See the just mentioned link, p. 8. This limit is derived from the limit of G(n,x)/n with G(n,x) given in the formula section.

Thanks go to Clyde P. Kruskal for asking a question which led to this comment.

(End)

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.

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.

See the general array and triangle for the average number of round trips of length L on (open) Laguerre graphs L_N given in A201198. Here a(n) = w(4,L=n), n>=0, the fourth row in this array. In the corresponding triangle this is the column no. N=4 without leading zeros: a(n) = A201198(n+3,4), n>=0.

For a sketch of this Laguerre graph L_4 see Figure 3 of the W. Lang link. The o.g.f. is also given there.

By definition the number of zero length round trips of length 0 for a vertex is put to 1 in order to count vertices.

For the general array and triangle for the total number of round trips of length L on closed Laguerre graphs Lc_N see A201199. Here a(n)=w(4,L=n)/4, n>=0, the fourth row of this array divided by 4. In the corresponding triangle a(n) = A201199(n+3,4)/4, n>=0.

For a sketch of the closed Laguerre graph Lc_4 see Figure 4 of the given W. Lang link. The o.g.f. is also found there.

By definition the number of length 0 round trips for a vertex is put to 1 in order to count vertices.

The average number of round trips of length n on a closed Laguerre graph Lc_N is in general a fraction. Therefore A201199 tabulates the total number of round trips.