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A377657 Array read by ascending antidiagonals: A(n, k) = Sum_{j=0..k} tan(j*Pi/(1 + 2*k))^(2*n).

%I A377657 #23 Nov 13 2024 17:22:00
%S A377657 1,0,2,0,3,3,0,9,10,4,0,27,90,21,5,0,81,850,371,36,6,0,243,8050,7077,
%T A377657 1044,55,7,0,729,76250,135779,33300,2365,78,8,0,2187,722250,2606261,
%U A377657 1070244,113311,4654,105,9,0,6561,6841250,50028755,34420356,5476405,312390,8295,136,10
%N A377657 Array read by ascending antidiagonals: A(n, k) = Sum_{j=0..k} tan(j*Pi/(1 + 2*k))^(2*n).
%C A377657 Based on an observation made by _Fredrik Johansson_ about A376777, which is the main diagonal of this array.
%F A377657 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
%e A377657 Array begins
%e A377657   [0] 1,    2,       3,         4,           5,            6, ... A000027
%e A377657   [1] 0,    3,      10,        21,          36,           55, ... A014105
%e A377657   [2] 0,    9,      90,       371,        1044,         2365, ... A377858
%e A377657   [3] 0,   27,     850,      7077,       33300,       113311, ... A376778
%e A377657   [4] 0,   81,    8050,    135779,     1070244,      5476405, ...
%e A377657   [5] 0,  243,   76250,   2606261,    34420356,    264893255, ...
%e A377657   [6] 0,  729,  722250,  50028755,  1107069876,  12813875437, ...
%e A377657   [7] 0, 2187, 6841250, 960335173, 35607151476, 619859803695, ...
%e A377657   .
%e A377657 Seen as a triangle T(n, k) = A(n-k, k):
%e A377657   [0] 1;
%e A377657   [1] 0,    2;
%e A377657   [2] 0,    3,      3;
%e A377657   [3] 0,    9,     10,       4;
%e A377657   [4] 0,   27,     90,      21,       5;
%e A377657   [5] 0,   81,    850,     371,      36,      6;
%e A377657   [6] 0,  243,   8050,    7077,    1044,     55,    7;
%e A377657   [7] 0,  729,  76250,  135779,   33300,   2365,   78,   8;
%e A377657   [8] 0, 2187, 722250, 2606261, 1070244, 113311, 4654, 105, 9;
%p A377657 A := (n, k) -> add(tan(j*Pi/(1 + 2*k))^(2*n), j = 0..k):
%p A377657 seq(print(seq(round(evalf(A(n, k), 32)), k = 0..6)), n = 0..7);
%o A377657 (PARI)
%o A377657 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
%Y A377657 Rows: A000027, A014105, A377858, A376778.
%Y A377657 Columns: A376478.
%Y A377657 Cf. A376777 (main diagonal), A377658 (antidiagonal sums).
%Y A377657 Cf. A091042.
%K A377657 nonn,tabl
%O A377657 0,3
%A A377657 _Peter Luschny_, Nov 10 2024