login
A209706
Triangle of coefficients of polynomials v(n,x) jointly generated with A209705; see the Formula section.
3
1, 3, 2, 4, 7, 4, 5, 14, 18, 8, 6, 23, 46, 44, 16, 7, 34, 92, 136, 104, 32, 8, 47, 160, 320, 376, 240, 64, 9, 62, 254, 640, 1016, 992, 544, 128, 10, 79, 378, 1148, 2296, 3024, 2528, 1216, 256, 11, 98, 536, 1904, 4592, 7616, 8576, 6272, 2688, 512, 12, 119
OFFSET
1,2
COMMENTS
Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x) = x*u(n-1,x)+x*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x)+(x+1)v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
T(n,k) = 2*T(n-1,k)+2*T(n-1,k-1)-T(n-2,k)-2*T(n-2,k-1), T(1,0)=1, T(2,0)=3, T(2,1)=2, T(3,0)=4, T(3,1)=7, T(3,2)=4, T(n,k)=0 if k<0 or if k>=n. - Philippe Deléham, Dec 27 2013
EXAMPLE
First five rows:
1
3...2
4...7....4
5...14...18...8
6...23...46...44...16
First three polynomials v(n,x): 1, 3 + 2x , 4 + 7x + 4x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%] (* A209705 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209706 *)
CROSSREFS
Sequence in context: A123097 A368218 A352419 * A350077 A134571 A054086
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 12 2012
STATUS
approved