A210230 Triangle of coefficients of polynomials v(n,x) jointly generated with A210229; see the Formula section.
1, 3, 1, 6, 4, 1, 11, 11, 5, 1, 19, 26, 17, 6, 1, 32, 56, 48, 24, 7, 1, 53, 114, 121, 78, 32, 8, 1, 87, 223, 283, 223, 117, 41, 9, 1, 142, 424, 627, 584, 372, 166, 51, 10, 1, 231, 789, 1334, 1434, 1073, 579, 226, 62, 11, 1, 375, 1444, 2750, 3352, 2879, 1818
Offset: 1
Examples
First five rows: 1 3....1 6....4....1 11...11...5....1 19...26...17...6...1 First three polynomials v(n,x): 1, 3 + x , 6 + 4x + x^2.
Programs
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Mathematica
u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1; v[n_, x_] := (x + 1)*u[n - 1, 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[%] (* A210229 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A210230 *)
Formula
u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
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