A164942 - OEIS

%I #32 Sep 08 2022 08:45:47

%S 1,3,-1,9,-6,1,27,-27,9,-1,81,-108,54,-12,1,243,-405,270,-90,15,-1,

%T 729,-1458,1215,-540,135,-18,1,2187,-5103,5103,-2835,945,-189,21,-1,

%U 6561,-17496,20412,-13608,5670,-1512,252,-24,1,19683,-59049,78732,-61236,30618,-10206,2268,-324,27,-1

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

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

%C 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

%C Row n: expansion of (3-x)^n. - _Philippe Deléham_, Oct 09 2011

%C Essentially the same as the inverse of A027465, but with opposite signs in every other row. - _M. F. Hasler_, Feb 17 2020

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

%H G. C. Greubel, <a href="/A164942/b164942.txt">Rows n = 0..100 of triangle, flattened</a>

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

%F T(n,k) = 3*T(n-1,k) - T(n-1,k-1). - _Philippe Deléham_, Oct 09 2011

%F G.f.: 1/(1-3*x+x*y). - _R. J. Mathar_, Aug 11 2015

%e Begins as triangle:

%e     1;

%e     3,   -1;

%e     9,   -6,    1;

%e    27,  -27,    9,   -1;

%e    81, -108,   54,  -12,    1;

%e   243, -405,  270,  -90,   15,   -1;

%p seq(seq( (-1)^k*binomial(n, k)*3^(n-k), k=0..n), n=0..10); # _G. C. Greubel_, Feb 17 2020

%t 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 *)

%o (Magma) [(-1)^k*Binomial(n, k)*3^(n-k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 17 2020

%o (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

%Y Cf. A000079, A001906, A027465.

%K sign,tabl

%O 0,2

%A _Mark Dols_, Sep 01 2009

%E More terms from _Philippe Deléham_, Oct 09 2011

%E a(46) corrected by _Georg Fischer_, Feb 17 2020

%E Title changed by _G. C. Greubel_, Feb 17 2020