A173590 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, 3, 3, 4, 10, 4, 5, 31, 31, 5, 6, 102, 182, 102, 6, 7, 367, 1093, 1093, 367, 7, 8, 1402, 8032, 8738, 8032, 1402, 8, 9, 5511, 67763, 86181, 86181, 67763, 5511, 9, 10, 21910, 600322, 1166470, 813802, 1166470, 600322, 21910, 10, 11, 87463, 5385001, 18015797, 11900131, 11900131, 18015797, 5385001, 87463, 11

Offset: 0

Author

Roger L. Bagula, Feb 22 2010

Keywords

• nonn
• easy
• tabl

Examples

			Square array begins:
  n\k | 1    2      3        4         5          6 ...
  -----------------------------------------------------
    0 | 2    3      4        5         6          7 ...
    1 | 3   10     31      102       367       1402 ...
    2 | 4   31    182     1093      8032      67763 ...
    3 | 5  102   1093     8738     86181    1166470 ...
    4 | 6  367   8032    86181    813802   11900131 ...
    5 | 7 1402  67763  1166470  11900131  124387562 ...
    6 | 8 5511 600322 18015797 260198052 2527336267 ...
    ...
		

Crossrefs

Cf. A173588, A173591.

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}];
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

T(n,k) = A173588(n,k) + A173588(k-1,n+1).

Extensions

Edited by Franck Maminirina Ramaharo, Jan 24 2019