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URL: https://oeis.org/A193817

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A193817
Mirror of the triangle A193816.
2
1, 2, 1, 6, 5, 1, 14, 17, 7, 1, 30, 49, 31, 9, 1, 62, 129, 111, 49, 11, 1, 126, 321, 351, 209, 71, 13, 1, 254, 769, 1023, 769, 351, 97, 15, 1, 510, 1793, 2815, 2561, 1471, 545, 127, 17, 1, 1022, 4097, 7423, 7937, 5503, 2561, 799, 161, 19, 1, 2046, 9217, 18943
OFFSET
0,2
COMMENTS
A193817 is obtained by reversing the rows of the triangle A193816.
Triangle T(n,k), read by rows, given by (2,1,-2,2,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined by A084938. - Philippe Deléham, Oct 05 2011
FORMULA
Write w(n,k) for the triangle at A193816. The triangle at A193817 is then given by w(n,n-k).
T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 2*T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(2,0)=6, T(2,1)=5, T(2,2)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Dec 15 2013
G.f.: (1-x+2*x^2+x^2*y)/((x-1)*(-1+2*x+x*y)). - R. J. Mathar, Aug 12 2015
EXAMPLE
First six rows:
1;
2, 1;
6, 5, 1;
14, 17, 7, 1;
30, 49, 31, 9, 1;
62, 129, 111, 49, 11, 1;
MATHEMATICA
z = 10; c = 1; d = 2;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193816 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A193817 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Aug 06 2011
STATUS
approved