A164942 Triangle, read by rows, T(n,k) = (-1)^k*binomial(n, k)*3^(n-k).

1, 3, -1, 9, -6, 1, 27, -27, 9, -1, 81, -108, 54, -12, 1, 243, -405, 270, -90, 15, -1, 729, -1458, 1215, -540, 135, -18, 1, 2187, -5103, 5103, -2835, 945, -189, 21, -1, 6561, -17496, 20412, -13608, 5670, -1512, 252, -24, 1, 19683, -59049, 78732, -61236, 30618, -10206, 2268, -324, 27, -1

Offset

0,2

Comments

Rows sum up to A000079, antidiagonals sum up to A001906.

Triangle, read by rows, given by [3,0,0,0,0,0,0,0,...] DELTA [ -1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 02 2009

Row n: expansion of (3-x)^n. - Philippe Deléham, Oct 09 2011

Essentially the same as the inverse of A027465, but with opposite signs in every other row. - M. F. Hasler, Feb 17 2020

The inverse of A027465 is (-1)^(n-k)*binomial(n, k)*3^(n - k). - G. C. Greubel, Feb 17 2020

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(n,k) = (-1)^n*(Inverse of A027465).

T(n,k) = 3*T(n-1,k) - T(n-1,k-1). - Philippe Deléham, Oct 09 2011

G.f.: 1/(1-3*x+x*y). - R. J. Mathar, Aug 11 2015

Example

Begins as triangle:
    1;
    3,   -1;
    9,   -6,    1;
   27,  -27,    9,   -1;
   81, -108,   54,  -12,    1;
  243, -405,  270,  -90,   15,   -1;

Maple

seq(seq( (-1)^k*binomial(n, k)*3^(n-k), k=0..n), n=0..10); # G. C. Greubel, Feb 17 2020

Mathematica

With[{m = 9}, CoefficientList[CoefficientList[Series[1/(1-3*x+x*y), {x, 0, m}, {y, 0, m}], x], y]] // Flatten (* Georg Fischer, Feb 17 2020 *)

Prog

(Magma) [(-1)^k*Binomial(n, k)*3^(n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 17 2020
(Sage) [[(-1)^k*binomial(n, k)*3^(n-k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 17 2020

Crossrefs

Cf. A000079, A001906, A027465.

Sequence in context: A105545 A178831 A027465 * A236420 A187537 A246256

Adjacent sequences: A164939 A164940 A164941 * A164943 A164944 A164945

Keyword

sign,tabl

Author

Mark Dols, Sep 01 2009

Extensions

More terms from Philippe Deléham, Oct 09 2011

a(46) corrected by Georg Fischer, Feb 17 2020

Title changed by G. C. Greubel, Feb 17 2020

Status

approved