A209235 -id:A209235 - cp's OEIS frontend

A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)(-1)^rbinomial(2q-r,r)x^r) / sum_{s=0,...,q} ((-1)^sbinomial(2q-s,s)*x^s), where q=1,2,....

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

Offset: 0

L. Edson Jeffery, Jan 23 2012

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

			Array begins as
1,  1,  1,  1,   1,    1, ...
2,  3,  7, 18,  47,  123, ...
3,  5, 13, 38, 117,  370, ...
4,  7, 19, 58, 187,  622, ...
5,  9, 25, 78, 257,  874, ...
6, 11, 31, 98, 327, 1126, ...
...

S. Barbero, Dickson Polynomials, Chebyshev Polynomials, and Some Conjectures of Jeffery, Journal of Integer Sequences, 17 (2014), #14.3.8.

Conjecture. Transpose of array A186740.
Conjecture. Rows 0,1,2 (up to an offset) are A000012, A005248, A198636 (proved, see the Barbero, et al., reference there).
Conjecture. Columns 0,1,2,3,4 (up to an offset) are A000027, A005408, A016921, A114698, A114646.
Cf. A209235.

Conjecture. The n-th entry in row q is given by R_q(n) = 2^(2*n)*(sum_{j=1,...,n+1} (cos(j*Pi/(2*q+1)))^(2*n)), q >= 1, n >= 0.
Conjecture. G.f. for column n is of the form G_n(x) = H_n(x)/(1-x)^2, where H_n(x) is a polynomial in x, n >= 0.
Conjecture. 2*A185095(q,n) = A198632(2*q,n), q >= 1, n >= 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.

cp's OEIS Frontend

A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)(-1)^rbinomial(2q-r,r)x^r) / sum_{s=0,...,q} ((-1)^sbinomial(2q-s,s)*x^s), where q=1,2,....

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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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cp's OEIS Frontend

A185095 Rectangular array read by antidiagonals: row q has generating function 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), where q=1,2,....

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Examples

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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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Author

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Comments

Examples

Links

Crossrefs

Formula

A185095 Rectangular array read by antidiagonals: row q has generating function F_q(x) = sum_{r=0,...,q-1} ((q-r)(-1)^rbinomial(2q-r,r)x^r) / sum_{s=0,...,q} ((-1)^sbinomial(2q-s,s)*x^s), where q=1,2,....

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.