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

0,2

Comments

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

Links

Table of n, a(n) for n=0..44.

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

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).

Example

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)

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.

Sequence in context: A054335 A110681 A117852 * A080247 A078937 A167560

Adjacent sequences: A080242 A080243 A080244 * A080246 A080247 A080248

Keyword

sign,tabl

Author

Paul Barry, Feb 13 2003

Status

approved