A209578 Triangle of coefficients of polynomials v(n,x) jointly generated with A209577; see the Formula section.
1, 3, 1, 5, 3, 1, 9, 8, 4, 1, 15, 19, 13, 5, 1, 25, 41, 36, 19, 6, 1, 41, 84, 90, 60, 26, 7, 1, 67, 165, 210, 169, 92, 34, 8, 1, 109, 315, 465, 439, 287, 133, 43, 9, 1, 177, 588, 990, 1073, 818, 454, 184, 53, 10, 1, 287, 1079, 2043, 2502, 2178, 1405, 681, 246
Offset: 1
Examples
First five rows: 1 1...1 3...2....1 5...6....3....1 9...13...10...4...1 First three polynomials v(n,x): 1, 1 + x , 3 + 2x + 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]; 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[%] (* A209577 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A209578 *)
Formula
u(n,x)=x*u(n-1,x)+v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Comments