A185095 -id:A185095 - cp's OEIS frontend

A198632 Triangle version of the array of the number of closed paths of even length on the graph P_n (n vertices, n-1 edges).

1, 0, 2, 0, 2, 3, 0, 2, 4, 4, 0, 2, 8, 6, 5, 0, 2, 16, 14, 8, 6, 0, 2, 32, 36, 20, 10, 7, 0, 2, 64, 94, 56, 26, 12, 8, 0, 2, 128, 246, 164, 76, 32, 14, 9, 0, 2, 256, 644, 488, 234, 96, 38, 16, 10, 0, 2, 512, 1686, 1460, 740, 304, 116, 44, 18, 11, 0, 2, 1024, 4414, 4376, 2372, 992, 374, 136, 50, 20, 12, 0, 2, 2048, 11556, 13124, 7654, 3296, 1244, 444, 156, 56, 22, 13

Offset: 0

Wolfdieter Lang, Nov 02 2011

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)

			The array w(n,2*k) is
n\k  0  1   2   3   4    5    6     7     8      9 ...
1:   1  0   0   0   0    0    0     0     0      0
2:   2  2   2   2   2    2    2     2     2      2
3:   3  4   8  16  32   64  128   256   512   1024
4:   4  6  14  36  94  246  644  1686  4414  11556
5:   5  8  20  56 164  488 1460  4376 13124  39368
6:   6 10  26  76 234  740 2372  7654 24778  80338
7:   7 12  32  96 304  992 3296 11072 37440 127104
8:   8 14  38 116 374 1244 4220 14504 50294 175454
9:   9 16  44 136 444 1496 5144 17936 63164 224056
...
The triangle is
k\n 1  2    3    4    5    6   7    8   9 10 11 12 ...
0:  1
1:  0  2
2:  0  2    3
3:  0  2    4    4
4:  0  2    8    6    5
5:  0  2   16   14    8    6
6:  0  2   32   36   20   10   7
7:  0  2   64   94   56   26  12    8
8:  0  2  128  246  164   76  32   14   9
9:  0  2  256  644  488  234  96   38  16 10
10: 0  2  512 1686 1460  740 304  116  44 18 11
11: 0  2 1024 4414 4376 2372 992  374 136 50 20 12
...
n=3, l=2*k = 4: graph P_3 as 1-2-3, with eight walks of length 4, namely 12121, 12321, 21212, 23232, 21232, 23212, 32323 and 32123.

S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.
S. Barbero, U. Cerruti, N. Murru, Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials, J. Integer Seq., Vol. 16 (2013), Article 13.8.1.
Carlos M. da Fonseca, M. Lawrence Glasser, Victor Kowalenko, Basic trigonometric power sums with applications, arXiv:1601.07839 [math.NT], 2016. See Theorem 5.1.
W. Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math., 89 (1998) 237-256.

Column sequences: A000007, 2*A000012, A198633, 2*A005248, A198635, ...

a(k,n)=w(n,2*(k-n+2)), the total number of closed walks (paths) of length 2*(k-n+2) on the graph P_n, which looks like o-o-o...-o, with n vertices (nodes) and n-1 edges (lines), k+1>=n>=1.
O.g.f. G(n,x) for w(n,l), which vanishes for odd l, is
((n+1)*coth((n+1)*log((2*x)/(1-sqrt(1-(2*x)^2)))) - 1/sqrt(1-(2*x)^2))/sqrt(1-(2*x)^2). See the comment above for a version with Chebyshev S-polynomials.
Conjecture: For the array w(n,2*k) in the example below, w(2*q,2*k)/2 = A185095(q,k), q >= 1, k >= 0. - L. Edson Jeffery, Nov 23 2013

A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2n)(Sum_{j=1..n+1} (cos(jPi/(2q+1)))^(2*n)), n >= 0, q >= 1.

Original entry on oeis.org

1, 1, 2, 1, 3, 3, 1, 7, 5, 4, 1, 18, 13, 7, 5, 1, 47, 38, 19, 9, 6, 1, 123, 117, 58, 25, 11, 7, 1, 322, 370, 187, 78, 31, 13, 8, 1, 843, 1186, 622, 257, 98, 37, 15, 9, 1, 2207, 3827, 2110, 874, 327, 118, 43, 17, 10, 1, 5778, 12389, 7252, 3034, 1126, 397, 138, 49, 19, 11

Offset: 0

L. Edson Jeffery, Jan 21 2012

Row indices n begin with 0, column indices q begin with 1.

			Array begins:
1    2     3     4     5     6     7     8     9 ...
1    3     5     7     9    11    13    15    17 ...
1    7    13    19    25    31    37    43    49 ...
1   18    38    58    78    98   118   138   158 ...
1   47   117   187   257   327   397   467   537 ...
1  123   370   622   874  1126  1378  1630  1882 ...
1  322  1186  2110  3034  3958  4882  5806  6730 ...
1  843  3827  7252 10684 14116 17548 20980 24412 ...
1 2207 12389 25147 38017 50887 63757 76627 89497 ...
...
As a triangle:
1,
1,  2,
1,  3,  3,
1,  7,  5,  4,
1, 18, 13,  7, 5,
1, 47, 38, 19, 9, 6,
...

Andrew Howroyd, Table of n, a(n) for n = 0..1275
S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.

Conjecture: Transpose of array A185095.
Conjecture: Columns 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).
Conjecture: Rows 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.
Cf. A209235.

Conjecture: G.f. for column q is F_q(x) = (Sum_{r=0..q-1} ((q-r)*(-1)^r*binomial(2*q-r,r)*x^r)) / (Sum_{s=0..q} ((-1)^s*binomial(2*q-s,s)*x^s)), q >= 1.

Conjecture: G.f. for n-th row is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x.

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 1, 3, 1, 5, 1, 5, 4, 1, 6, 1, 7, 4, 7, 1, 8, 1, 11, 4, 19, 16, 18, 1, 9, 1, 13, 4, 25, 16, 38, 29, 1, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 12, 1, 19, 4, 43, 16, 98, 64, 187, 193, 123, 1

Offset: 1

L. Edson Jeffery, Jan 12 2013

Antidiagonal sums: {1,3,5,9,16,26,46,78,136,...}.

			Array begins as
.1..1...1..1...1...1
.2..1...3..4...7..11
.3..1...5..4..13..16
.4..1...7..4..19..16
.5..1...9..4..25..16
.6..1..11..4..31..16

Rows: cf. A000012, A000032, A094649, A189234, A216605, etc.

Cf. A185095, A186740.

T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.
Empirical g.f. for row n: F(x) = (Sum_{u=0..n-1} A122765(n,n-1-u)*x^u)/(Sum_{v=0..n} A108299(n,v)*x^v).
Empirical: odd column first differences tend to A000984 = {1, 2, 6, 20, 70, 252, ...} (central binomial coefficients).

A232441 Sequence read from antidiagonals of rectangular array given by A(n,k) = 2^(2k)(Sum_{j=1..n-floor(n/2)-1} (cos(jPi/n))^(2k)), rows n >= 3, columns k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 4, 1, 3, 4, 7, 8, 1, 3, 5, 10, 18, 16, 1, 4, 6, 13, 28, 47, 32, 1, 4, 7, 16, 38, 82, 123, 64, 1, 5, 8, 19, 48, 117, 244, 322, 128, 1, 5, 9, 22, 58, 152, 370, 730, 843, 256, 1, 6, 10, 25, 68

Offset: 3

L. Edson Jeffery, Nov 23 2013

Row indices n begin with 3, column indices k begin with 0.

			1,    1,    1,    1,    1,    1,    1,    1,    1,    1,    1,...
1,    2,    4,    8,   16,   32,   64,  128,  256,  512, 1024,...
2,    3,    7,   18,   47,  123,  322,  843, 2207, 5778,15127,...
2,    4,   10,   28,   82,  244,  730, 2188, 6562,19684,59050,...

Cf. A186740, A185095, A198632, A198636.

Table[Function[m, FullSimplify[2^(2 k)*Sum[Cos[j*Pi/m]^(2 k), {j, m - Floor[m/2] - 1}]]][n - k + 1], {n, 3, 12}, {k, 0, n - 2}] // Flatten (* Michael De Vlieger, Mar 18 2017 *)

A(2*m+1,k) = A186740(m,k), m = 1,2,....

Conjecture: A(n,k) = floor(A198632(n-1,k)/2), n >= 3, k >= 0.

cp's OEIS Frontend

A198632 Triangle version of the array of the number of closed paths of even length on the graph P_n (n vertices, n-1 edges).

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A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2n)(Sum_{j=1..n+1} (cos(jPi/(2q+1)))^(2*n)), n >= 0, q >= 1.

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A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.

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A232441 Sequence read from antidiagonals of rectangular array given by A(n,k) = 2^(2k)(Sum_{j=1..n-floor(n/2)-1} (cos(jPi/n))^(2k)), rows n >= 3, columns k >= 0.

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cp's OEIS Frontend

A198632 Triangle version of the array of the number of closed paths of even length on the graph P_n (n vertices, n-1 edges).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2*n)*(Sum_{j=1..n+1} (cos(j*Pi/(2*q+1)))^(2*n)), n >= 0, q >= 1.

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Examples

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Formula

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}*Sum_{j=1..n} (cos((2*j-1)*Pi/(2*n+1)))^{k-1}.

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Formula

A232441 Sequence read from antidiagonals of rectangular array given by A(n,k) = 2^(2*k)*(Sum_{j=1..n-floor(n/2)-1} (cos(j*Pi/n))^(2*k)), rows n >= 3, columns k >= 0.

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Author

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Comments

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Programs

Mathematica

Formula

A186740 Sequence read from antidiagonals of rectangular array with entry in row n and column q given by T(n,q) = 2^(2n)(Sum_{j=1..n+1} (cos(jPi/(2q+1)))^(2*n)), n >= 0, q >= 1.

A209235 Rectangular array read by antidiagonals, with entry k in row n given by T(n,k) = 2^{k-1}Sum_{j=1..n} (cos((2j-1)Pi/(2n+1)))^{k-1}.

A232441 Sequence read from antidiagonals of rectangular array given by A(n,k) = 2^(2k)(Sum_{j=1..n-floor(n/2)-1} (cos(jPi/n))^(2k)), rows n >= 3, columns k >= 0.