A173588 T(n,k) = (k^n)*U(n, (1/k + k)/2), where U(n,x) is the n-th Chebyshev polynomial of the second kind, square array read by antidiagonals upward (n >= 0, k >= 1).
1, 2, 1, 3, 5, 1, 4, 21, 10, 1, 5, 85, 91, 17, 1, 6, 341, 820, 273, 26, 1, 7, 1365, 7381, 4369, 651, 37, 1, 8, 5461, 66430, 69905, 16276, 1333, 50, 1, 9, 21845, 597871, 1118481, 406901, 47989, 2451, 65, 1, 10, 87381, 5380840, 17895697, 10172526, 1727605, 120100, 4161, 82, 1
Offset: 0
Examples
Square array begins: n\k | 1 2 3 4 5 6 ... ----------------------------------------------------- 0 | 1 1 1 1 1 1 ... 1 | 2 5 10 17 26 37 ... 2 | 3 21 91 273 651 1333 ... 3 | 4 85 820 4369 16276 47989 ... 4 | 5 341 7381 69905 406901 1727605 ... 5 | 6 1365 66430 1118481 10172526 62193781 ... 6 | 7 5461 597871 17895697 254313151 2238976117 ... ...
Programs
-
Mathematica
p[x_, q_] = 1/(x^2 - (1/q + q)*x + 1); a = Table[Table[n^m*SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], {m, 0, 20}], {n, 1, 21}]; Flatten[Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]] -
Maxima
T(n, k) := k^n*chebyshev_u(n, (1/k + k)/2)$ create_list(T(n - k + 1, k), n, 0, 12, k, 1, n + 1); /* Franck Maminirina Ramaharo, Jan 18 2019 */
Formula
T(n,k) = (k^n)*([x^n] 1/(x^2 - (1/k + k)*x + 1)).
Extensions
Edited by Franck Maminirina Ramaharo, Jan 24 2019
Comments