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

%I A352362 #22 Feb 16 2025 08:34:03
%S A352362 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,
%T A352362 76,119,82,18,0,2,7,38,140,322,393,198,29,2,2,8,51,234,727,1364,1298,
%U A352362 478,47,0,2,9,66,364,1442,3775,5778,4287,1154,76,2
%N A352362 Array read by ascending antidiagonals. T(n, k) = L(k, n) where L are the Lucas polynomials.
%H A352362 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasPolynomial.html">Lucas Polynomial</a>
%F A352362 T(n, k) = Sum_{j=0..floor(k/2)} binomial(k-j, j)*(k/(k-j))*n^(k-2*j) for k >= 1.
%F A352362 T(n, k) = (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k.
%F A352362 T(n, k) = [x^k] ((2 - n*x)/(1 - n*x - x^2)).
%F A352362 T(n, k) = n^k*hypergeom([1/2 - k/2, -k/2], [1 - k], -4/n^2) for n,k >= 1.
%e A352362 Array starts:
%e A352362 n\k 0, 1,  2,   3,    4,     5,      6,       7,        8, ...
%e A352362 --------------------------------------------------------------
%e A352362 [0] 2, 0,  2,   0,    2,     0,      2,       0,        2, ... A010673
%e A352362 [1] 2, 1,  3,   4,    7,    11,     18,      29,       47, ... A000032
%e A352362 [2] 2, 2,  6,  14,   34,    82,    198,     478,     1154, ... A002203
%e A352362 [3] 2, 3, 11,  36,  119,   393,   1298,    4287,    14159, ... A006497
%e A352362 [4] 2, 4, 18,  76,  322,  1364,   5778,   24476,   103682, ... A014448
%e A352362 [5] 2, 5, 27, 140,  727,  3775,  19602,  101785,   528527, ... A087130
%e A352362 [6] 2, 6, 38, 234, 1442,  8886,  54758,  337434,  2079362, ... A085447
%e A352362 [7] 2, 7, 51, 364, 2599, 18557, 132498,  946043,  6754799, ... A086902
%e A352362 [8] 2, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, ... A086594
%e A352362 [9] 2, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, ... A087798
%e A352362 A007395|A059100|
%e A352362     A001477 A079908
%p A352362 T := (n, k) -> (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k:
%p A352362 seq(seq(simplify(T(n - k, k)), k = 0..n), n = 0..10);
%t A352362 Table[LucasL[k, n], {n, 0, 9}, {k, 0, 9}] // TableForm
%t A352362 (* or *)
%t A352362 T[ 0, k_] := 2 Mod[k+1, 2]; T[n_, 0] := 2;
%t A352362 T[n_, k_] := n^k Hypergeometric2F1[1/2 - k/2, -k/2, 1 - k, -4/n^2];
%t A352362 Table[T[n, k], {n, 0, 9}, {k, 0, 8}] // TableForm
%o A352362 (PARI)
%o A352362 T(n, k) = ([0, 1; 1, k]^n*[2; k])[1, 1] ;
%o A352362 export(T)
%o A352362 for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))
%Y A352362 Rows: A010673, A000032, A002203, A006497, A014448, A087130, A085447, A086902, A086594, A087798.
%Y A352362 Columns: A007395, A001477, A059100, A079908.
%Y A352362 Cf. A320570 (main diagonal), A114525, A309220 (variant), A117938 (variant), A352361 (Fibonacci polynomials), A350470 (Jacobsthal polynomials).
%K A352362 nonn,tabl
%O A352362 0,1
%A A352362 _Peter Luschny_, Mar 18 2022