A206022 Riordan array (1, x*exp(arcsinh(-2*x))).

1, 0, 1, 0, -2, 1, 0, 2, -4, 1, 0, 0, 8, -6, 1, 0, -2, -8, 18, -8, 1, 0, 0, 0, -32, 32, -10, 1, 0, 4, 8, 30, -80, 50, -12, 1, 0, 0, 0, 0, 128, -160, 72, -14, 1, 0, -10, -16, -28, -112, 350, -280, 98, -16, 1, 0, 0, 0

Offset

0,5

Comments

Riordan array (1, x*(sqrt(1+4x^2)-2x)); inverse of Riordan array (1, x/sqrt(1-4x)), see A205813.

The g.f. for row sums (1,1,-1,-1,3,1,-9,1,27,13,-81,67,243,...) is (1+2*x^2+x*sqrt(1+4*x^2))/(1+3*x^2).

Triangle T(n,k), read by rows, given by (0, -2, 1, -1, 1, -1, 1, -1, 1, -1, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Links

Table of n, a(n) for n=0..57.

Formula

T(n,n) = 1, T(n+1,n) = -2n = -A005843(n), T(n+2,n) = 2*n^2 = A001105(n), T(n+3,n) = -A130809(n+1), T(2n,n) = A009117(n), T(2n+3,1) = (-1)^n*2*A000108(n).

T(n,k) = T(n-2,k-2) - 4*T(n-2,k-1), for k >= 2.

Example

Triangle begins:
  1
  0,   1
  0,  -2,   1
  0,   2,  -4,   1
  0,   0,   8,  -6,    1,
  0,  -2,  -8,  18,   -8,    1
  0,   0,   0, -32,   32,  -10,     1
  0,   4,   8,  30,  -80,   50,   -12,    1
  0,   0,   0,   0,  128, -160,    72,  -14,    1
  0, -10, -16, -28, -112,  350,  -280,   98,  -16,   1
  0,   0,   0,   0,    0, -512,   768, -448,  128, -18,   1
  0,  28,  40,  54,   96,  420, -1512, 1470, -672, 162, -20, 1

Crossrefs

Cf. A104624 (column k=1).

Sequence in context: A061009 A144106 A104558 * A115247 A204163 A122542

Adjacent sequences: A206019 A206020 A206021 * A206023 A206024 A206025

Keyword

easy,sign,tabl

Author

Philippe Deléham, Feb 02 2012

Status

approved