A126124 Triangle, matrix inverse of A124733, companion to A123965.

1, -2, 1, 5, -5, 1, -13, 19, -8, 1, 34, -65, 42, -11, 1, -89, 210, -183, 74, -14, 1, 233, -654, 717, -394, 115, -17, 1, -610, 1985, -2622, 1825, -725, 165, -20, 1, 1597, -5911, 9134, -7703, 3885, -1203, 224, -23, 1

Offset: 1

Author

Gary W. Adamson, Dec 17 2006

Keywords

• tabl
• sign

Comments

Left border (unsigned) = odd-indexed Fibonacci numbers. Left border (unsigned) of A123965 = even-indexed Fibonacci numbers.

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

Equals A129818*A130595 as lower triangular matrices. - Philippe Deléham, Oct 26 2007

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

Apart from signs, equals A123971.

Matrix inverse of A124733.

Examples

			First few rows of the triangle are:
    1;
   -2,    1;
    5,   -5,    1;
  -13,   19,   -8,    1;
   34,  -65,   42,  -11,    1;
  -89,  210, -183,   74,  -14,    1;
  ...
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:
  1;
  0,    1;
  0,   -2,    1;
  0,    5,   -5,    1;
  0,  -13,   19,   -8,    1;
  0,   34,  -65,   42,  -11,    1;
  0,  -89,  210, -183,   74,  -14,    1;
  0,  233, -654,  717, -394,  115,  -17,    1;
		

Crossrefs

Cf. A123965, A124733, A123971, A152063.

Formula

Sum_{k=1..n} (-1)^(n-k)*T(n,k) = A001835(n). - Philippe Deléham, Jul 14 2007

T(n,k) = T(n-1,k-1) - 3*T(n-1,k) - T(n-2,k). - Philippe Deléham, Dec 13 2011

T(n,k) = (-1)^(n+k)*Sum_{m=k..n} binomial(m,k)*binomial(m+n,2*m). - Wadim Zudilin, Jan 11 2012

G.f.: (1+x)*x*y/(1+3*x+x^2-x*y). - R. J. Mathar, Aug 11 2015

Extensions

Corrected by Philippe Deléham, Jul 14 2007

More terms from Philippe Deléham, Dec 13 2011