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 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

%I

%S 2,1,1,0,-6,0,1,-3,-3,1,2,-18,110,-18,2,1,-35,-159,-159,-35,1,0,10,

%T 3000,-5790,3000,10,0,1,-139,-15091,27457,27457,-15091,-139,1,2,30,

%U 110454,-595250,578402,-595250,110454,30,2,1,5,-715167,7576241,-5255603,7576241,-715167,5,1

%N 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).

%F Let b(n,k) = (k^n)*U(n, (1/k - k)/2). Then T(n,k) = b(n,k) + b(k-1,n+1).

%e Square array begins:

%e n\k | 1 2 3 4 5 6 ...

%e ----------------------------------------------------

%e 0 | 2 1 0 1 2 1 ...

%e 1 | 1 -6 -3 -18 -35 10 ...

%e 2 | 0 -3 110 -159 3000 -15091 ...

%e 3 | 1 -18 -159 -5790 27457 -595250 ...

%e 4 | 2 -35 3000 27457 578402 -5255603 ...

%e 5 | 1 10 -15091 -595250 -5255603 -92967910 ...

%e 6 | 0 -139 110454 7576241 156747480 1344158389 ...

%e ...

%t p[x_, q_] = 1/(x^2 - (1/q - q)*x + 1);

%t a = Table[Table[n^m*SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], {m, 0, 20}], {n, 1, 21}];

%t b = (a + Transpose[a]);

%t Flatten[Table[Table[b[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]]

%o (Maxima)

%o 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)\$

%o create_list(T(n - k + 1, k), n, 0, 12, k, 1, n + 1);

%o /* _Franck Maminirina Ramaharo_, Jan 24 2019 */

%Y Cf. A173588, A173590.

%K sign,easy,tabl

%O 0,1

%A _Roger L. Bagula_, Feb 22 2010

%E Edited by _Franck Maminirina Ramaharo_, Jan 24 2019

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Last modified May 7 23:48 EDT 2021. Contains 343652 sequences. (Running on oeis4.)