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A210287
Triangle of coefficients of polynomials v(n,x) jointly generated with A209999; see the Formula section.
3
1, 3, 1, 6, 6, 1, 11, 18, 10, 1, 19, 45, 41, 15, 1, 32, 100, 130, 80, 21, 1, 53, 208, 352, 310, 141, 28, 1, 87, 413, 866, 994, 652, 231, 36, 1, 142, 794, 1991, 2828, 2429, 1253, 358, 45, 1, 231, 1490, 4358, 7391, 7871, 5348, 2248, 531, 55, 1, 375, 2745
OFFSET
1,2
COMMENTS
Column 1: -2+F(n+3), where F=000045 (Fibonacci numbers)
Row sums: A003462
Alternating row sums: 1,2,1,2,1,2,1,2,1,2,1,2,1,2,...
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)+1,
v(n,x)=u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
3....1
6....6....1
11...18...10...1
19...45...41...15...1
First three polynomials v(n,x): 1, 3 + x , 6 + 6x + x^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] + 1;
v[n_, x_] := 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[%] (* A209999 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210287 *)
CROSSREFS
Sequence in context: A325005 A325013 A152685 * A116412 A089511 A246257
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 23 2012
STATUS
approved