login
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,....
5
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
0,2
COMMENTS
Row indices q begin with 1, column indices n begin with 0.
LINKS
FORMULA
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
EXAMPLE
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, ...
...
CROSSREFS
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.
Sequence in context: A125175 A210552 A193376 * A177888 A073020 A090349
KEYWORD
nonn,tabl
AUTHOR
L. Edson Jeffery, Jan 23 2012
STATUS
approved