A352362 Array read by ascending antidiagonals. T(n, k) = L(k, n) where L are the Lucas polynomials.

2, 2, 0, 2, 1, 2, 2, 2, 3, 0, 2, 3, 6, 4, 2, 2, 4, 11, 14, 7, 0, 2, 5, 18, 36, 34, 11, 2, 2, 6, 27, 76, 119, 82, 18, 0, 2, 7, 38, 140, 322, 393, 198, 29, 2, 2, 8, 51, 234, 727, 1364, 1298, 478, 47, 0, 2, 9, 66, 364, 1442, 3775, 5778, 4287, 1154, 76, 2

Offset

0,1

Links

Table of n, a(n) for n=0..65.

Eric Weisstein's World of Mathematics, Lucas Polynomial

Formula

T(n, k) = Sum_{j=0..floor(k/2)} binomial(k-j, j)*(k/(k-j))*n^(k-2*j) for k >= 1.

T(n, k) = (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k.

T(n, k) = [x^k] ((2 - n*x)/(1 - n*x - x^2)).

T(n, k) = n^k*hypergeom([1/2 - k/2, -k/2], [1 - k], -4/n^2) for n,k >= 1.

Example

Array starts:
n\k 0, 1,  2,   3,    4,     5,      6,       7,        8, ...
--------------------------------------------------------------
[0] 2, 0,  2,   0,    2,     0,      2,       0,        2, ... A010673
[1] 2, 1,  3,   4,    7,    11,     18,      29,       47, ... A000032
[2] 2, 2,  6,  14,   34,    82,    198,     478,     1154, ... A002203
[3] 2, 3, 11,  36,  119,   393,   1298,    4287,    14159, ... A006497
[4] 2, 4, 18,  76,  322,  1364,   5778,   24476,   103682, ... A014448
[5] 2, 5, 27, 140,  727,  3775,  19602,  101785,   528527, ... A087130
[6] 2, 6, 38, 234, 1442,  8886,  54758,  337434,  2079362, ... A085447
[7] 2, 7, 51, 364, 2599, 18557, 132498,  946043,  6754799, ... A086902
[8] 2, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, ... A086594
[9] 2, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, ... A087798
A007395|A059100|
    A001477 A079908

Maple

T := (n, k) -> (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k:
seq(seq(simplify(T(n - k, k)), k = 0..n), n = 0..10);

Mathematica

Table[LucasL[k, n], {n, 0, 9}, {k, 0, 9}] // TableForm
(* or *)
T[ 0, k_] := 2 Mod[k+1, 2]; T[n_, 0] := 2;
T[n_, k_] := n^k Hypergeometric2F1[1/2 - k/2, -k/2, 1 - k, -4/n^2];
Table[T[n, k], {n, 0, 9}, {k, 0, 8}] // TableForm

Prog

(PARI)
T(n, k) = ([0, 1; 1, k]^n*[2; k])[1, 1] ; export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))

Crossrefs

Rows: A010673, A000032, A002203, A006497, A014448, A087130, A085447, A086902, A086594, A087798.

Columns: A007395, A001477, A059100, A079908.

Cf. A320570 (main diagonal), A114525, A309220 (variant), A117938 (variant), A352361 (Fibonacci polynomials), A350470 (Jacobsthal polynomials).

Sequence in context: A050949 A074943 A272011 * A236374 A045719 A114906

Adjacent sequences: A352359 A352360 A352361 * A352363 A352364 A352365

Keyword

nonn,tabl

Author

Peter Luschny, Mar 18 2022

Status

approved