|
%I #24 Dec 30 2023 23:54:18
%S 1,1,-1,3,-4,1,9,-15,7,-1,27,-54,36,-10,1,81,-189,162,-66,13,-1,243,
%T -648,675,-360,105,-16,1,729,-2187,2673,-1755,675,-153,19,-1,2187,
%U -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
%N Fibonacci matrix read by antidiagonals. (Inverse of A136158.)
%C 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
%H G. C. Greubel, <a href="/A164948/b164948.txt">Rows n = 0..50 of the triangle, flattened</a>
%F Sum_{k=0..n} T(n, k) = A000007(n).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A001519(n).
%F From _Philippe Deléham_, Oct 09 2011: (Start)
%F 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.
%F Row n: Expansion of (1-x)*(3-x)^(n-1), n>0. (End)
%F G.f.: (1-2*x)/(1-3*x+x*y). - _R. J. Mathar_, Aug 12 2015
%F From _G. C. Greubel_, Dec 26 2023: (Start)
%F T(n, k) = (-1)^k * A136158(n, k).
%F T(n, k) = (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k), for n > 0, with T(0, 0) = 1.
%F T(n, 0) = A133494(n).
%F T(n, 1) = -A006234(n+2), n >= 1.
%F T(n, 2) = A080420(n-2), n >= 2.
%F T(n, 3) = -A080421(n-3), n >= 3.
%F T(2*n, n) = 4*(-1)^n*A098399(n-1) - (1/3)*[n=0].
%F T(n, n-4) = 27*(-1)^n*A001296(n-3), n >= 4.
%F T(n, n-3) = 9*(-1)^(n-1)*A002411(n-2), n >= 3.
%F T(n, n-2) = 3*(-1)^n*A000326(n-1) = (-1)^n*A062741(n-1), n >= 2.
%F T(n, n-1) = (-1)^(n-1)*A016777(n-1), n >= 1.
%F T(n, n) = (-1)^n.
%F Sum_{k=0..n} (-1)^k*T(n, k) = A081294(n).
%F Sum_{k=0..n} (-1)^k*T(n-k, k) = A003688(n). (End)
%e As triangle:
%e 1;
%e 1, -1;
%e 3, -4, 1;
%e 9, -15, 7, -1;
%e 27, -54, 36, -10, 1;
%e 81, -189, 162, -66, 13, -1;
%e 243, -648, 675, -360, 105, -16, 1;
%t A164948[n_,k_]:= If[n==0,1,(-1)^k*3^(n-k-1)*(n+2*k)*Binomial[n,k]/n];
%t Table[A164948[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Dec 26 2023 *)
%o (Magma)
%o A164948:= func< n,k | n eq 0 select 1 else (-1)^k*3^(n-k-1)*(n+2*k)*Binomial(n,k)/n >;
%o [A164948(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Dec 26 2023
%o (SageMath)
%o def A164948(n,k): return 1 if (n==0) else (-1)^k*3^(n-k-1)*((n+2*k)/n)*binomial(n, k)
%o flatten([[A164948(n, k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Dec 26 2023
%Y Cf. A000007, A000326, A001296, A001519, A002411, A003688, A006234.
%Y Cf. A016777, A038763, A080420, A080421, A081294, A084938, A098399.
%Y Cf. A133494, A136158, A164942.
%K tabl,sign
%O 0,4
%A _Mark Dols_, Sep 01 2009
%E More terms from _Philippe Deléham_, Oct 09 2011
|
