OFFSET
1,3
COMMENTS
For n > 1, n-th alternating row sum = (-1)^(n-1)*F(2*n-3), where F=A000045 (Fibonacci numbers).
Coefficient of x^(n-1) in u(n,x): 2^(n-1).
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 11 2012
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
u(n,x) = x*u(n-1,x) + v(n-1,x),
v(n,x) = u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 11 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(2,0) = 1, T(2,1) = 2, T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+x-3*y*x-y*x^2+2*y^2*x^2)/(1-3*y*x-(1-2y^2)*x^2). (End)
EXAMPLE
First five rows:
1;
1, 2;
1, 3, 4;
1, 5, 7, 8;
1, 6, 17, 15, 16;
First three polynomials v(n,x):
1
1 + 2x
1 + 3x + 4x^2.
From Philippe Deléham, Mar 11 2012: (Start)
(1, 0, -1/2, -1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 3, 4, 0;
1, 5, 7, 8, 0;
1, 6, 17, 15, 16, 0;
1, 8, 23, 49, 31, 32, 0;
1, 9, 39, 72, 129, 63, 64, 0;
1, 11, 48, 150, 201, 321, 127, 128, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A209172 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209413 *)
CoefficientList[CoefficientList[Series[(1 + x - 3*y*x - y*x^2 + 2*y^2*x^2)/(1 - 3*y*x - (1 - 2 y^2)*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 08 2012
STATUS
approved