A173591 T(n, k) = k^n*U(n, (1/k - k)/2) + (n + 1)^(k - 1)*U(k - 1, (1/(n + 1) - n - 1)/2), where U(n,x) is the n-th Chebyshev polynomial of the second kind, square array read by antidiagonals (n >= 0, k >= 1).
2, 1, 1, 0, -6, 0, 1, -3, -3, 1, 2, -18, 110, -18, 2, 1, -35, -159, -159, -35, 1, 0, 10, 3000, -5790, 3000, 10, 0, 1, -139, -15091, 27457, 27457, -15091, -139, 1, 2, 30, 110454, -595250, 578402, -595250, 110454, 30, 2, 1, 5, -715167, 7576241, -5255603, 7576241, -715167, 5, 1
Offset: 0
Examples
Square array begins:
n\k | 1 2 3 4 5 6 ...
----------------------------------------------------
0 | 2 1 0 1 2 1 ...
1 | 1 -6 -3 -18 -35 10 ...
2 | 0 -3 110 -159 3000 -15091 ...
3 | 1 -18 -159 -5790 27457 -595250 ...
4 | 2 -35 3000 27457 578402 -5255603 ...
5 | 1 10 -15091 -595250 -5255603 -92967910 ...
6 | 0 -139 110454 7576241 156747480 1344158389 ...
...
Programs
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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}]; b = (a + Transpose[a]); Flatten[Table[Table[b[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]] -
Maxima
T(n, k) := k^n*chebyshev_u(n, (1/k - k)/2) + (n + 1)^(k - 1)*chebyshev_u(k - 1, (1/(n + 1) - n - 1)/2)$ create_list(T(n - k + 1, k), n, 0, 12, k, 1, n + 1); /* Franck Maminirina Ramaharo, Jan 24 2019 */
Formula
Let b(n,k) = (k^n)*U(n, (1/k - k)/2). Then T(n,k) = b(n,k) + b(k-1,n+1).
Extensions
Edited by Franck Maminirina Ramaharo, Jan 24 2019