A075117 Table by antidiagonals of generalized Lucas numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=2 and T(n,1)=1.

2, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 4, 5, 1, 2, 1, 7, 7, 7, 1, 2, 1, 11, 17, 10, 9, 1, 2, 1, 18, 31, 31, 13, 11, 1, 2, 1, 29, 65, 61, 49, 16, 13, 1, 2, 1, 47, 127, 154, 101, 71, 19, 15, 1, 2, 1, 76, 257, 337, 297, 151, 97, 22, 17, 1, 2, 1, 123, 511, 799, 701, 506, 211, 127, 25, 19, 1, 2

Offset: 0

Author

Henry Bottomley, Sep 02 2002

Keywords

• nonn
• tabl

Examples

			Array starts as:
  2, 1,  1,  1,  1,   1, ...;
  2, 1,  3,  4,  7,  11, ...;
  2, 1,  5,  7, 17,  31, ...;
  2, 1,  7, 10, 31,  61, ...;
  2, 1,  9, 13, 49, 101, ...;
  2, 1, 11, 16, 71, 151, ...; etc.
		

Links

• G. C. Greubel, Antidiagonals n = 0..100, flattened

Crossrefs

Cf. A060959.

Rows include: A054977, A000032, A014551, A075118, A072265.

Columns include: A007395, A000012, A005408, A016777, A056220, A062786.

Programs

Magma

[2^(1+k-n)*(&+[Binomial(n-k,2*j)*(1+4*k)^j: j in [0..Floor((n-k)/2)]]): k in [0..n], n in [0..13]]; // G. C. Greubel, Jan 27 2020

Maple

seq(seq( 2^(1+k-n)*add( binomial(n-k, 2*j)*(1+4*k)^j, j=0..floor((n-k)/2)), k=0..n), n=0..13); # G. C. Greubel, Jan 27 2020

Mathematica

T[n_, k_]:= ((1 + Sqrt[1+4n])/2)^k + ((1 - Sqrt[1+4n])/2)^k; Table[If[n==0 && k==0, 2, T[k, n-k]]//Simplify, {n,0,13}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 27 2020 *)

Sage

def T(n, k): return 2^(1-k)*sum( binomial(k, 2*j)*(1+4*n)^j for j in (0..floor(k/2)) )
[[T(k,n-k) for k in (0..n)] for n in (0..13)] # G. C. Greubel, Jan 27 2020

Formula

T(n, k) = ((1+sqrt(4*n+1))/2)^k + ((1-sqrt(4*n+1))/2)^k = 2*A060959(n, k+1) - A060959(n, k).

T(n, k) = 2^(1-k)*Sum_{j=0..floor(k/2)} binomial(k, 2*j)*(1+4*n)^j. - G. C. Greubel, Jan 27 2020