A209418 Triangle of coefficients of polynomials v(n,x) jointly generated with A209417; see the Formula section.
1, 1, 3, 1, 4, 7, 1, 7, 13, 15, 1, 8, 30, 38, 31, 1, 11, 42, 104, 103, 63, 1, 12, 69, 178, 321, 264, 127, 1, 15, 87, 331, 657, 921, 649, 255, 1, 16, 124, 484, 1354, 2200, 2512, 1546, 511, 1, 19, 148, 760, 2266, 4978, 6856, 6598, 3595, 1023, 1, 20, 195, 1020, 3870, 9384, 16938, 20226, 16827, 8204, 2047
Offset: 1
Examples
First five rows: 1; 1, 3; 1, 4, 7; 1, 7, 13, 15; 1, 8, 30, 38, 31; First three polynomials v(n,x): 1 1 + 3x 1 + 4x + 7x^2. From _Philippe Deléham_, Apr 01 2012: (Start) (1, 0, -2/3, -1/3, 0, 0, 0, ...) DELTA (0, 3, -2/3, 2/3, 0, 0, 0, ...) begins: 1; 1, 0; 1, 3, 0; 1, 4, 7, 0; 1, 7, 13, 15, 0; 1, 8, 30, 38, 31, 0; (End)
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Programs
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Mathematica
u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := x*u[n - 1, x] + v[n - 1, x]; v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*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[%] (* A209417 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A209418 *) CoefficientList[CoefficientList[Series[(1 + x)/(1 - 3*y*x - x^2 - y*x^2 + 2*y^2*x^2), {x,0,10}, {y,0,10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)
Formula
u(n,x) = x*u(n-1,x) + v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Apr 01 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1+x-3*y*x-y*x^2+2*y^2*x^2)/(1-3*y*x-x^2-y*x^2+2*y^2*x^2).
T(n,k) = 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) -2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
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