A210595 Triangle of coefficients of polynomials v(n,x) jointly generated with A209999; see the Formula section.

1, 2, 1, 3, 3, 2, 4, 6, 7, 3, 5, 10, 16, 13, 5, 6, 15, 30, 35, 25, 8, 7, 21, 50, 75, 76, 46, 13, 8, 28, 77, 140, 181, 157, 84, 21, 9, 36, 112, 238, 371, 413, 317, 151, 34, 10, 45, 156, 378, 686, 924, 911, 625, 269, 55, 11, 55, 210, 570, 1176, 1848, 2206, 1949, 1211, 475, 89

Offset: 1

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

Row n starts with n and ends with F(n), where F=A000045 (Fibonacci numbers).

Row sums: A048739.

Alternating row sums: 1,1,2,2,3,3,4,4,5,5, ...

For a discussion and guide to related arrays, see A208510.

Examples

			First few rows are:
  1;
  2,  1;
  3,  3,  2;
  4,  6,  7,  3;
  5, 10, 16, 13,  5;
  6, 15, 30, 35, 25,  8;
  7, 21, 50, 75, 76, 46, 13;
First few polynomials v(n,x) are:
  v(1, x) = 1;
  v(2, x) = 2 +  1*x;
  v(3, x) = 3 +  3*x +  2*x^2;
  v(4, x) = 4 +  6*x +  7*x^2 +  3*x^3;
  v(5, x) = 5 + 10*x + 16*x^2 + 13*x^3 + 5*x^4;
		

Links

• G. C. Greubel, Rows n = 1..30 of the triangle, flattened

Crossrefs

Cf. A208510, A210565.

Programs

Mathematica

(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 16;
u[n_, x_]:= x*u[n-1, x] + (1+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[%]    (* 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 *)
v[n_, x_]:= v[n, x]= If[n<2, n+1 +n*x, (1+x)*v[n-1, x] +x^2*v[n-2, x] +1];
T[n_]:= CoefficientList[Series[v[n, x], {x,0,n}], x];
Table[T[n-1], {n, 12}]//Flatten (* G. C. Greubel, May 24 2021 *)

Sage

@CachedFunction
def v(n,x): return n+1+n*x if (n<2) else (1+x)*v(n-1,x) +x^2*v(n-2,x) +1
def T(n): return taylor( v(n,x) , x,0,n).coefficients(x, sparse=False)
flatten([T(n-1) for n in (1..12)]) # G. C. Greubel, May 24 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]( v(n,x) ), where v(n, x) = (1+x)*v(n-1, x) + x^2*v(n-2, x) + 1, v(1, x) = 1, and v(2, x) = 2 + x. - G. C. Greubel, May 24 2021