A209750 Triangle of coefficients of polynomials v(n,x) jointly generated with A209749; see the Formula section.
1, 2, 1, 2, 4, 1, 3, 6, 7, 1, 3, 11, 15, 11, 1, 4, 14, 32, 32, 16, 1, 4, 21, 51, 79, 61, 22, 1, 5, 25, 84, 152, 174, 107, 29, 1, 5, 34, 118, 277, 393, 352, 176, 37, 1, 6, 39, 172, 447, 796, 915, 666, 275, 46, 1, 6, 50, 225, 705, 1446, 2060, 1965, 1193, 412, 56, 1
Offset: 1
Examples
First five rows: 1 2 1 2 4 1 3 6 7 1 3 11 15 11 1 First three polynomials v(n,x): 1, 2 + x, 2 + 4*x + 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] + (x + 1)*v[n - 1, x]; v[n_, x_] := u[n - 1, x] + 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[%] (* A209749 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A209750 *)
Formula
u(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Comments