A080245 Inverse of coordination sequence array A113413.

1, -2, 1, 6, -4, 1, -22, 16, -6, 1, 90, -68, 30, -8, 1, -394, 304, -146, 48, -10, 1, 1806, -1412, 714, -264, 70, -12, 1, -8558, 6752, -3534, 1408, -430, 96, -14, 1, 41586, -33028, 17718, -7432, 2490, -652, 126, -16, 1

Offset: 0

Author

Paul Barry, Feb 13 2003

Keywords

• sign
• tabl

Comments

Formal inverse of A035607 when written as lower triangular matrix 1 2 1 2 4 1 ...

Examples

			Rows are {1}, {-2, 1}, {6, -4, 1}, {-22, 16, -6, 1}, ....
From _Paul Barry_, Apr 28 2009: (Start)
Triangle begins
  1,
  -2, 1,
  6, -4, 1,
  -22, 16, -6, 1,
  90, -68, 30, -8, 1,
  -394, 304, -146, 48, -10, 1,
  1806, -1412, 714, -264, 70, -12, 1
Production matrix is
  -2, 1,
  2, -2, 1,
  -2, 2, -2, 1,
  2, -2, 2, -2, 1,
  -2, 2, -2, 2, -2, 1,
  2, -2, 2, -2, 2, -2, 1,
  -2, 2, -2, 2, -2, 2, -2, 1 (End)
		

Links

• Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 7.

Crossrefs

Row sums are signed little Schroeder numbers A080243. Diagonal sums are given by A080244.

Cf. A035607, A080243, A080244, A006603, A001003.

Cf. A000007 A084938.

Essentially same triangle as A033877 but with rows read in reversed order.

Formula

Essentially the same as the triangle T(n, k), for n>0 and k>0, given by [0, -2, -1, -2, -1, -2, -1, -2, ...] DELTA A000007. Triangle (unsigned) given by [0, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007, where DELTA is Deléham's operator defined in A084938.

Riordan array ((sqrt(1+6x+x^2)-x-1)/(2x), (sqrt(1+6x+x^2)-x-1)/2).