A209745 Triangle of coefficients of polynomials u(n,x) jointly generated with A209746; see the Formula section.
1, 1, 2, 2, 5, 4, 3, 12, 16, 8, 5, 25, 49, 44, 16, 8, 50, 127, 166, 112, 32, 13, 96, 301, 513, 504, 272, 64, 21, 180, 670, 1408, 1808, 1424, 640, 128, 34, 331, 1427, 3562, 5641, 5816, 3824, 1472, 256, 55, 600, 2939, 8494, 15981, 20330, 17520, 9888
Offset: 1
Examples
First five rows: 1 1...2 2...5....4 3...12...16...8 5...25...49...44...16 First three polynomials u(n,x): 1, 1 + 2x, 2 + 5x + 4x^2. (0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 1, 0, 0, 0, ...) begins : 1 0, 1 0, 1, 2 0, 2, 5, 4 0, 3, 12, 16, 8 0, 5, 25, 49, 44, 16 ... - _Philippe Deléham_, Mar 24 2012
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) = T(2,0) = 1, T(2,1) = 2, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 24 2012
Comments