A210549 Triangle of coefficients of polynomials u(n,x) jointly generated with A210550; see the Formula section.
1, 1, 2, 1, 3, 4, 1, 3, 9, 8, 1, 3, 10, 25, 16, 1, 3, 10, 33, 65, 32, 1, 3, 10, 34, 104, 161, 64, 1, 3, 10, 34, 115, 311, 385, 128, 1, 3, 10, 34, 116, 381, 886, 897, 256, 1, 3, 10, 34, 116, 395, 1224, 2421, 2049, 512, 1, 3, 10, 34, 116, 396, 1334, 3796, 6388
Offset: 1
Examples
First five rows: 1 1...2 1...3...4 1...3...9....8 1...3...10...25...16 First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 4x^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] + (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[%] (* A210235 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A210236 *)
Formula
u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
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