OFFSET
0,3
COMMENTS
Based on an observation made by Fredrik Johansson about A376777, which is the main diagonal of this array.
FORMULA
Row n of A091042(n, k) = binomial(2*n+1, 2*k) gives the polynomial Pe(n, x), with zeros in -tan(Pi/2*n+1)^2, -tan(2*Pi/2*n+1)^2, ..., -tan(n*Pi/2*n+1)^2. Let Pm(n, k, x) be the polynomial with zeros in (-tan(Pi/2*n+1)^2)^k, (-tan(2*Pi/2*n+1)^2)^k, ..., (-tan(n*Pi/2*n+1)^2)^k, then A(k, n) is the coefficient of X^(n-1) in the polynomial Pm(n, k, x). A way to do this calculation without evaluation of irrational numbers is to obtain the companion matrix M of the polynomial Pe(n, x), then A(k, n) = tr(M^k) (the trace of M^k). - Thomas Scheuerle, Nov 11 2024
EXAMPLE
Array begins
[0] 1, 2, 3, 4, 5, 6, ... A000027
[1] 0, 3, 10, 21, 36, 55, ... A014105
[2] 0, 9, 90, 371, 1044, 2365, ... A377858
[3] 0, 27, 850, 7077, 33300, 113311, ... A376778
[4] 0, 81, 8050, 135779, 1070244, 5476405, ...
[5] 0, 243, 76250, 2606261, 34420356, 264893255, ...
[6] 0, 729, 722250, 50028755, 1107069876, 12813875437, ...
[7] 0, 2187, 6841250, 960335173, 35607151476, 619859803695, ...
.
Seen as a triangle T(n, k) = A(n-k, k):
[0] 1;
[1] 0, 2;
[2] 0, 3, 3;
[3] 0, 9, 10, 4;
[4] 0, 27, 90, 21, 5;
[5] 0, 81, 850, 371, 36, 6;
[6] 0, 243, 8050, 7077, 1044, 55, 7;
[7] 0, 729, 76250, 135779, 33300, 2365, 78, 8;
[8] 0, 2187, 722250, 2606261, 1070244, 113311, 4654, 105, 9;
MAPLE
A := (n, k) -> add(tan(j*Pi/(1 + 2*k))^(2*n), j = 0..k):
seq(print(seq(round(evalf(A(n, k), 32)), k = 0..6)), n = 0..7);
PROG
(PARI)
A(n, k) = {trace(matcompanion(sum(m=0, k, x^m*binomial(2*k+1, 2*(k-m))*(-1)^(m+1)))^n)+(n==0) } \\ Thomas Scheuerle, Nov 11 2024
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Nov 10 2024
STATUS
approved