%I #13 Mar 20 2022 10:13:01
%S 1,1,1,1,1,1,1,1,3,1,1,1,5,5,1,1,1,7,9,11,1,1,1,9,13,29,21,1,1,1,11,
%T 17,55,65,43,1,1,1,13,21,89,133,181,85,1,1,1,15,25,131,225,463,441,
%U 171,1,1,1,17,29,181,341,937,1261,1165,341,1
%N Array read by ascending antidiagonals. T(n, k) = J(k, n) where J are the Jacobsthal polynomials.
%F T(n, k) = Sum_{j=0..k} binomial(k - j, j)*(2*n)^j.
%F T(n, k) = ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s) where s = sqrt(8*n + 1).
%F T(n, k) = [x^k] (1 / (1 - x - 2*n*x^2)).
%F T(n, k) = hypergeom([1/2 - k/2, -k/2], [-k], -8*n).
%e Array starts:
%e n\k 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
%e ---------------------------------------------------------------------
%e [0] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
%e [1] 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... A001045
%e [2] 1, 1, 5, 9, 29, 65, 181, 441, 1165, 2929, ... A006131
%e [3] 1, 1, 7, 13, 55, 133, 463, 1261, 4039, 11605, ... A015441
%e [4] 1, 1, 9, 17, 89, 225, 937, 2737, 10233, 32129, ... A015443
%e [5] 1, 1, 11, 21, 131, 341, 1651, 5061, 21571, 72181, ... A015446
%e [6] 1, 1, 13, 25, 181, 481, 2653, 8425, 40261, 141361, ... A053404
%e [7] 1, 1, 15, 29, 239, 645, 3991, 13021, 68895, 251189, ... A350468
%e [8] 1, 1, 17, 33, 305, 833, 5713, 19041, 110449, 415105, ... A168579
%e [9] 1, 1, 19, 37, 379, 1045, 7867, 26677, 168283, 648469, ... A350469
%e A005408 | A082108 |
%e A016813 A014641
%p J := (n, x) -> add(2^k*binomial(n - k, k)*x^k, k = 0..n):
%p seq(seq(J(k, n-k), k = 0..n), n = 0..10);
%t T[n_, k_] := Hypergeometric2F1[(1 - k)/2, -k/2, -k, -8 n];
%t Table[T[n, k], {n, 0, 9}, {k, 0, 9}] // TableForm
%t (* or *)
%t T[n_, k_] := With[{s = Sqrt[8*n+1]}, ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s)];
%t Table[Simplify[T[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm
%o (PARI)
%o T(n, k) = ([1, 2; k, 0]^n)[1, 1] ; export(T)
%o for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))
%Y Rows: A000012, A001045, A006131, A015441, A015443, A015446, A053404, A350468, A168579, A350469.
%Y Columns: A000012, A005408, A016813, A082108, A014641.
%Y Cf. A350467 (main diagonal), A352361 (Fibonacci polynomials), A352362 (Lucas polynomials).
%K nonn,tabl
%O 0,9
%A _Peter Luschny_, Mar 19 2022
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