A208609 Triangle of coefficients of polynomials v(n,x) jointly generated with A208608; see the Formula section.
1, 2, 2, 2, 4, 3, 2, 6, 9, 5, 2, 8, 17, 18, 8, 2, 10, 27, 41, 35, 13, 2, 12, 39, 76, 93, 66, 21, 2, 14, 53, 125, 196, 200, 122, 34, 2, 16, 69, 190, 360, 472, 415, 222, 55, 2, 18, 87, 273, 603, 957, 1083, 837, 399, 89, 2, 20, 107, 376, 945, 1750, 2400, 2392
Offset: 1
Examples
First five rows: 1 2...2 2...4...3 2...6...9....5 2...8...17...18...8 First five polynomials v(n,x): 1 2 + 2x 2 + 4x + 3x^2 2 + 6x + 9x^2 + 5x^3 2 + 8x + 17x^2 + 18x^3 + 8x^4
Crossrefs
Cf. A208608.
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_] := (x + 1)*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[%] (* A208608 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A208609 *)
Formula
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Comments