A164948 Fibonacci matrix read by antidiagonals. (Inverse of A136158.)

1, 1, -1, 3, -4, 1, 9, -15, 7, -1, 27, -54, 36, -10, 1, 81, -189, 162, -66, 13, -1, 243, -648, 675, -360, 105, -16, 1, 729, -2187, 2673, -1755, 675, -153, 19, -1, 2187, -7290, 10206, -7938, 3780, -1134, 210, -22, 1, 6561, -24057, 37908, -34020, 19278, -7182, 1764, -276, 25, -1, 19683, -78732, 137781, -139968, 91854, -40824, 12474, -2592, 351, -28, 1

Offset: 0

Author

Mark Dols, Sep 01 2009

Keywords

• tabl
• sign

Comments

Triangle, read by rows, given by [1,2,0,0,0,0,0,0,0,...] DELTA [-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 02 2009

Examples

			As triangle:
    1;
    1,   -1;
    3,   -4,    1;
    9,  -15,    7,   -1;
   27,  -54,   36,  -10,    1;
   81, -189,  162,  -66,   13,   -1;
  243, -648,  675, -360,  105,  -16,    1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000007, A000326, A001296, A001519, A002411, A003688, A006234.

Cf. A016777, A038763, A080420, A080421, A081294, A084938, A098399.

Cf. A133494, A136158, A164942.

Programs

Magma

A164948:= func< n,k | n eq 0 select 1 else (-1)^k*3^(n-k-1)*(n+2*k)*Binomial(n,k)/n >;
[A164948(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 26 2023

Mathematica

A164948[n_,k_]:= If[n==0,1,(-1)^k*3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
Table[A164948[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 26 2023 *)

SageMath

def A164948(n,k): return 1 if (n==0) else (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
flatten([[A164948(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 26 2023

Formula

Sum_{k=0..n} T(n, k) = A000007(n).

Sum_{k=0..floor(n/2)} T(n-k, k) = A001519(n).

From Philippe Deléham, Oct 09 2011: (Start)

T(n,k) = 3*T(n-1,k) - T(n-1,k-1) with T(0,0)=1, T(1,0)=1, T(1,1)=-1.

Row n: Expansion of (1-x)*(3-x)^(n-1), n>0. (End)

G.f.: (1-2*x)/(1-3*x+x*y). - R. J. Mathar, Aug 12 2015

From G. C. Greubel, Dec 26 2023: (Start)

T(n, k) = (-1)^k * A136158(n, k).

T(n, k) = (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k), for n > 0, with T(0, 0) = 1.

T(n, 0) = A133494(n).

T(n, 1) = -A006234(n+2), n >= 1.

T(n, 2) = A080420(n-2), n >= 2.

T(n, 3) = -A080421(n-3), n >= 3.

T(2*n, n) = 4*(-1)^n*A098399(n-1) - (1/3)*[n=0].

T(n, n-4) = 27*(-1)^n*A001296(n-3), n >= 4.

T(n, n-3) = 9*(-1)^(n-1)*A002411(n-2), n >= 3.

T(n, n-2) = 3*(-1)^n*A000326(n-1) = (-1)^n*A062741(n-1), n >= 2.

T(n, n-1) = (-1)^(n-1)*A016777(n-1), n >= 1.

T(n, n) = (-1)^n.

Sum_{k=0..n} (-1)^k*T(n, k) = A081294(n).

Sum_{k=0..n} (-1)^k*T(n-k, k) = A003688(n). (End)

Extensions

More terms from Philippe Deléham, Oct 09 2011