A210565 Triangle of coefficients of polynomials u(n,x) jointly generated with A210595; see the Formula section.
1, 2, 2, 3, 5, 3, 4, 9, 10, 5, 5, 14, 22, 20, 8, 6, 20, 40, 51, 38, 13, 7, 27, 65, 105, 111, 71, 21, 8, 35, 98, 190, 256, 233, 130, 34, 9, 44, 140, 315, 511, 594, 474, 235, 55, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 11, 65, 255, 726, 1554, 2534, 3130, 2860, 1836, 744, 144
Offset: 1
Examples
First five rows: 1; 2, 2; 3, 5, 3; 4, 9, 10, 5; 5, 14, 22, 20, 8; First three polynomials u(n,x): u(1, x) = 1; u(2, x) = 2 + 2*x; u(3, x) = 3 + 5*x + 3*x^2.
Links
- G. C. Greubel, Rows n = 1..30 of the triangle, flattened
Programs
-
Mathematica
(* First program *) u[1, x_]:= 1; v[1, x_]:= 1; z = 16; u[n_, x_]:= x*u[n-1, x] + (x+1)*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[%] (* A210565 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A210595 *) (* Second program *) u[n_, x_]:= u[n, x]= If[n<2, (n+1)*(1+x)^n, (1+x)*u[n-1, x] +x^2*u[n-2, x] +1+x]; T[n_]:= CoefficientList[Series[u[n, x], {x, 0, n}], x]; Table[T[n-1], {n,12}] (* G. C. Greubel, May 23 2021 *) -
Sage
@CachedFunction def u(n,x): return (n+1)*(1+x)^n if (n<2) else (1+x)*u(n-1,x) + x^2*u(n-2,x) +1+x def T(n): return taylor( u(n,x) , x,0,n).coefficients(x, sparse=False) flatten([T(n-1) for n in (1..12)]) # G. C. Greubel, May 23 2021
Formula
u(n,x) = x*u(n-1,x) + (x+1)*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.
T(n, k) = [x^k]( u(n, x) ), where u(n, x) = (1+x)*u(n-1,x) + x^2*u(n-2,x) + 1 + x, u(1, x) = 1, and u(2, x) = 2 + 2*x. - G. C. Greubel, May 24 2021
Comments