A208521 Triangle of coefficients of polynomials v(n,x) jointly generated with A208520; see the Formula section.
1, 2, 2, 3, 4, 2, 4, 6, 8, 4, 5, 8, 20, 16, 4, 6, 10, 40, 40, 24, 8, 7, 12, 70, 80, 84, 48, 8, 8, 14, 112, 140, 224, 168, 64, 16, 9, 16, 168, 224, 504, 448, 288, 128, 16, 10, 18, 240, 336, 1008, 1008, 960, 576, 160, 32, 11, 20, 330, 480, 1848, 2016, 2640, 1920
Offset: 1
Examples
First five rows: 1 2...2 3...4...2 4...6...8....4 5...8...20...16...4 First five polynomials v(n,x): 1 2 + 2x 3 + 4x + 2x^2 4 + 6x + 8x^2 + 4x^3 5 + 8x + 20x^2 + 16x^3 + 4x^4
Crossrefs
Cf. A208520.
Programs
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Mathematica
u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; v[n_, x_] := 2 x*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[%] (* A208520 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A208521 *)
Formula
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=2x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Comments