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.

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

0,3

Comments

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

Links

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.

Formula

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.

Example

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,
...

Crossrefs

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.

Sequence in context: A284979 A127123 A271238 * A353585 A103525 A294432

Adjacent sequences: A186737 A186738 A186739 * A186741 A186742 A186743

Keyword

nonn,tabl

Author

L. Edson Jeffery, Jan 21 2012

Status

approved