login
A209751
Triangle of coefficients of polynomials u(n,x) jointly generated with A209752; see the Formula section.
3
1, 1, 2, 2, 5, 4, 2, 10, 15, 8, 3, 14, 35, 39, 16, 3, 22, 63, 108, 95, 32, 4, 27, 109, 235, 309, 223, 64, 4, 38, 160, 454, 782, 838, 511, 128, 5, 44, 242, 770, 1688, 2408, 2183, 1151, 256, 5, 58, 322, 1264, 3226, 5776, 7009, 5512, 2559, 512, 6, 65, 450
OFFSET
1,3
COMMENTS
Alternating row sums: 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+1)*v(n-1,x),
v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
1...2
2...5....4
2...10...15...8
3...14...35...39...16
First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 x*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[%] (* A209751 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209752 *)
CROSSREFS
Sequence in context: A068465 A217876 A209771 * A275381 A283235 A209763
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 14 2012
STATUS
approved