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

1,2

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.

Links

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

Cf. A208510, A210565.

Sequence in context: A332435 A361744 A324751 * A094435 A133341 A111492

Adjacent sequences: A210592 A210593 A210594 * A210596 A210597 A210598

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 23 2012

Status

approved