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%I #40 Jun 05 2021 08:41:08
%S 1,-2,1,5,-5,1,-13,19,-8,1,34,-65,42,-11,1,-89,210,-183,74,-14,1,233,
%T -654,717,-394,115,-17,1,-610,1985,-2622,1825,-725,165,-20,1,1597,
%U -5911,9134,-7703,3885,-1203,224,-23,1
%N Triangle, matrix inverse of A124733, companion to A123965.
%C Left border (unsigned) = odd-indexed Fibonacci numbers. Left border (unsigned) of A123965 = even-indexed Fibonacci numbers.
%C Subtriangle of the triangle T(n,k) given by [0,-2,-1/2,-1/2,0,0,0,0,...] DELTA [1,0,1/2,-1/2,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 02 2007
%C Equals A129818*A130595 as lower triangular matrices. - _Philippe Deléham_, Oct 26 2007
%C Reversals = bisection of triangle A152063: (1; 1,2; 1,5,5; ...) having the following property: Product_{k=1..floor((n-1)/2)} (1 + 4*cos^2 k*2Pi/n) = the odd-indexed Fibonacci numbers. Example: x^3 - 8x^2 - 19x + 13 relates to the heptagon, and with k=1,2,3,..., the product = 13. - _Gary W. Adamson_, Aug 15 2010
%C Apart from signs, equals A123971.
%C Matrix inverse of A124733.
%F Sum_{k=1..n} (-1)^(n-k)*T(n,k) = A001835(n). - _Philippe Deléham_, Jul 14 2007
%F T(n,k) = T(n-1,k-1) - 3*T(n-1,k) - T(n-2,k). - _Philippe Deléham_, Dec 13 2011
%F T(n,k) = (-1)^(n+k)*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - _Wadim Zudilin_, Jan 11 2012
%F G.f.: (1+x)*x*y/(1+3*x+x^2-x*y). - _R. J. Mathar_, Aug 11 2015
%e First few rows of the triangle are:
%e 1;
%e -2, 1;
%e 5, -5, 1;
%e -13, 19, -8, 1;
%e 34, -65, 42, -11, 1;
%e -89, 210, -183, 74, -14, 1;
%e ...
%e Triangle (n >= 0 and 0 <= k <= n) [0,-2,-1/2,-1/2,0,0,0,0,0,...] DELTA [1,0,1/2,-1/2,0,0,0,0,0,...] begins:
%e 1;
%e 0, 1;
%e 0, -2, 1;
%e 0, 5, -5, 1;
%e 0, -13, 19, -8, 1;
%e 0, 34, -65, 42, -11, 1;
%e 0, -89, 210, -183, 74, -14, 1;
%e 0, 233, -654, 717, -394, 115, -17, 1;
%Y Cf. A123965, A124733, A123971, A152063.
%K tabl,sign
%O 1,2
%A _Gary W. Adamson_, Dec 17 2006
%E Corrected by _Philippe Deléham_, Jul 14 2007
%E More terms from _Philippe Deléham_, Dec 13 2011
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