A202389 Triangle T(n,k), read by rows, given by (1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 1, 1, -1, 1, 2, -1, -2, 2, 3, 1, -2, -5, 3, 5, 1, 3, -5, -10, 5, 8, -1, 3, 9, -10, -20, 8, 13, -1, -4, 9, 22, -20, -38, 13, 21, 1, -4, -14, 22, 51, -38, -71, 21, 34, 1, 5, -14, -40, 51, 111, -71, -130, 34, 55

Offset

0,6

Comments

T(n,n) = A000045(n+1) = Fibonacci(n+1).

Links

Table of n, a(n) for n=0..54.

Formula

T(n,k) = T(n-1,k-1) + T(n-2,k-2) - T(n-2,k) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n<k.

G.f.: (1+x)/(1-y*x+(1-y^2)*x^2).

Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A046717(n), A000007(n), A057077(n), A040000(n), A000244(n) for x = -2, -1, 0, 1, 2 respectively.

Example

Triangle begins :
1
1, 1
-1, 1, 2
-1, -2, 2, 3
1, -2, -5, 3, 5
1, 3, -5, -10, 5, 8
-1, 3, 9, -10, -20, 8, 13

Mathematica

With[{m = 9}, CoefficientList[CoefficientList[Series[(1+x)/(1-y*x+(1-y^2)*x
^2), {x, 0 , m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

Prog

(PARI) T(n, k) = if (k<0, 0, if (n<k, 0, if ((k<=1) && (n<=1), 1, T(n-1, k-1) + T(n-2, k-2) - T(n-2, k))));
matrix(10, 10, n, k, T(n-1, k-1)) \\ to see the triangle \\ Michel Marcus, Feb 17 2020

Crossrefs

Cf. A000007, A000045, A000244, A040000, A046717, A057077, A084938, A124137, A123585.

Sequence in context: A109400 A370577 A333831 * A210868 A176853 A261787

Adjacent sequences: A202386 A202387 A202388 * A202390 A202391 A202392

Keyword

sign,tabl

Author

Philippe Deléham, Dec 18 2011

Extensions

a(52) corrected by Georg Fischer, Feb 17 2020

Status

approved