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

0,1

Links

Table of n, a(n) for n=0..53.

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).

Example

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 ...
    ...

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}]]

Prog

(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 */

Crossrefs

Cf. A173588, A173590.

Sequence in context: A238802 A229892 A064879 * A343320 A156603 A156612

Adjacent sequences: A173588 A173589 A173590 * A173592 A173593 A173594

Keyword

sign,easy,tabl

Author

Roger L. Bagula, Feb 22 2010

Extensions

Edited by Franck Maminirina Ramaharo, Jan 24 2019

Status

approved