A209746 Triangle of coefficients of polynomials v(n,x) jointly generated with A209745; see the Formula section.
1, 2, 2, 3, 7, 4, 5, 17, 20, 8, 8, 37, 65, 52, 16, 13, 75, 176, 210, 128, 32, 21, 146, 428, 679, 616, 304, 64, 34, 276, 971, 1921, 2312, 1696, 704, 128, 55, 511, 2097, 4970, 7449, 7240, 4464, 1600, 256, 89, 931, 4366, 12056, 21622, 26146, 21344
Offset: 1
Examples
First five rows: 1; 2, 2; 3, 7, 4; 5, 17, 20, 8; 8, 37, 65, 52, 16; First three polynomials v(n,x): 1 2 + 2x 3 + 7x + 4x^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_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x]; 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[%] (* A209745 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A209746 *)
Formula
u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(1,0) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k >= 0. - Philippe Deléham, Mar 24 2012
Comments