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

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

Row n starts with n and ends with F(n+1), where F=A000045 (Fibonacci numbers).

Row sums: A005409.

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0, ...

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

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

Crossrefs

Cf. A208510, A210595.

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