A210552 Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.
1, 1, 2, 1, 3, 3, 1, 4, 5, 5, 1, 5, 7, 10, 8, 1, 6, 9, 16, 18, 13, 1, 7, 11, 23, 31, 33, 21, 1, 8, 13, 31, 47, 62, 59, 34, 1, 9, 15, 40, 66, 101, 119, 105, 55, 1, 10, 17, 50, 88, 151, 205, 227, 185, 89, 1, 11, 19, 61, 113, 213, 321, 414, 426, 324, 144, 1, 12, 21, 73
Offset: 1
Examples
First five rows: 1 1...2 1...3...3 1...4...5...5 1...5...7...10...8 First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^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*v[n - 1, x] + 1; v[n_, x_] := 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[%] (* A210552 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A210553 *) Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *) Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *) Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *) Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)
Formula
u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
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