A350470 - OEIS

%I #17 Sep 30 2024 11:40:46

%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] ;

%o 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