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%I #15 Feb 28 2023 17:04:21
%S 1,-2,1,6,-4,1,-22,16,-6,1,90,-68,30,-8,1,-394,304,-146,48,-10,1,1806,
%T -1412,714,-264,70,-12,1,-8558,6752,-3534,1408,-430,96,-14,1,41586,
%U -33028,17718,-7432,2490,-652,126,-16,1
%N Inverse of coordination sequence array A113413.
%C Formal inverse of A035607 when written as lower triangular matrix 1 2 1 2 4 1 ...
%H Huyile Liang, Yanni Pei, and Yi Wang, <a href="https://arxiv.org/abs/2302.11856">Analytic combinatorics of coordination numbers of cubic lattices</a>, arXiv:2302.11856 [math.CO], 2023. See p. 7.
%F 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.
%F Riordan array ((sqrt(1+6x+x^2)-x-1)/(2x), (sqrt(1+6x+x^2)-x-1)/2).
%e Rows are {1}, {-2, 1}, {6, -4, 1}, {-22, 16, -6, 1}, ....
%e From _Paul Barry_, Apr 28 2009: (Start)
%e Triangle begins
%e 1,
%e -2, 1,
%e 6, -4, 1,
%e -22, 16, -6, 1,
%e 90, -68, 30, -8, 1,
%e -394, 304, -146, 48, -10, 1,
%e 1806, -1412, 714, -264, 70, -12, 1
%e Production matrix is
%e -2, 1,
%e 2, -2, 1,
%e -2, 2, -2, 1,
%e 2, -2, 2, -2, 1,
%e -2, 2, -2, 2, -2, 1,
%e 2, -2, 2, -2, 2, -2, 1,
%e -2, 2, -2, 2, -2, 2, -2, 1 (End)
%Y Row sums are signed little Schroeder numbers A080243. Diagonal sums are given by A080244.
%Y Cf. A035607, A080243, A080244, A006603, A001003.
%Y Cf. A000007 A084938.
%Y Essentially same triangle as A033877 but with rows read in reversed order.
%K sign,tabl
%O 0,2
%A _Paul Barry_, Feb 13 2003
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