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A350470 Array read by ascending antidiagonals. T(n, k) = J(k, n) where J are the Jacobsthal polynomials. 5
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, 17, 55, 65, 43, 1, 1, 1, 13, 21, 89, 133, 181, 85, 1, 1, 1, 15, 25, 131, 225, 463, 441, 171, 1, 1, 1, 17, 29, 181, 341, 937, 1261, 1165, 341, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
LINKS
FORMULA
T(n, k) = Sum_{j=0..k} binomial(k - j, j)*(2*n)^j.
T(n, k) = ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s) where s = sqrt(8*n + 1).
T(n, k) = [x^k] (1 / (1 - x - 2*n*x^2)).
T(n, k) = hypergeom([1/2 - k/2, -k/2], [-k], -8*n).
EXAMPLE
Array starts:
n\k 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
---------------------------------------------------------------------
[0] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
[1] 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, ... A001045
[2] 1, 1, 5, 9, 29, 65, 181, 441, 1165, 2929, ... A006131
[3] 1, 1, 7, 13, 55, 133, 463, 1261, 4039, 11605, ... A015441
[4] 1, 1, 9, 17, 89, 225, 937, 2737, 10233, 32129, ... A015443
[5] 1, 1, 11, 21, 131, 341, 1651, 5061, 21571, 72181, ... A015446
[6] 1, 1, 13, 25, 181, 481, 2653, 8425, 40261, 141361, ... A053404
[7] 1, 1, 15, 29, 239, 645, 3991, 13021, 68895, 251189, ... A350468
[8] 1, 1, 17, 33, 305, 833, 5713, 19041, 110449, 415105, ... A168579
[9] 1, 1, 19, 37, 379, 1045, 7867, 26677, 168283, 648469, ... A350469
MAPLE
J := (n, x) -> add(2^k*binomial(n - k, k)*x^k, k = 0..n):
seq(seq(J(k, n-k), k = 0..n), n = 0..10);
MATHEMATICA
T[n_, k_] := Hypergeometric2F1[(1 - k)/2, -k/2, -k, -8 n];
Table[T[n, k], {n, 0, 9}, {k, 0, 9}] // TableForm
(* or *)
T[n_, k_] := With[{s = Sqrt[8*n+1]}, ((1+s)^(k+1) - (1-s)^(k+1)) / (2^(k+1)*s)];
Table[Simplify[T[n, k]], {n, 0, 9}, {k, 0, 9}] // TableForm
PROG
(PARI)
T(n, k) = ([1, 2; k, 0]^n)[1, 1] ; export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))
CROSSREFS
Cf. A350467 (main diagonal), A352361 (Fibonacci polynomials), A352362 (Lucas polynomials).
Sequence in context: A254101 A349025 A348963 * A277604 A112475 A347232
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 19 2022
STATUS
approved

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Last modified March 4 02:53 EST 2024. Contains 370522 sequences. (Running on oeis4.)