A207611 Triangle of coefficients of polynomials v(n,x) jointly generated with A207610; see Formula section.

1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 8, 8, 5, 2, 1, 13, 15, 11, 6, 2, 1, 21, 28, 23, 14, 7, 2, 1, 34, 51, 47, 32, 17, 8, 2, 1, 55, 92, 93, 70, 42, 20, 9, 2, 1, 89, 164, 181, 148, 97, 53, 23, 10, 2, 1, 144, 290, 346, 306, 217, 128, 65, 26, 11, 2, 1, 233, 509, 653, 619, 472

Offset

1,2

Comments

Column 1: Fibonacci numbers, A000045

Column 2: A029907

Row sums: A003945.

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

Subtriangle of the triangle given by (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

Links

Table of n, a(n) for n=1..71.

Formula

u(n,x) = u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + x*v(n-1,x)+1, where u(1,x)=1, v(1,x)=1.

T(n,k) = T(n-1,k) + (n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(1,0) = T(2,1) = 1, T(2,0) = 2 and T(n,k) = 0 if k < 0 or if k >= n.

Example

First five rows:
  1;
  2, 1;
  3, 2, 1;
  5, 4, 2, 1;
  8, 8, 5, 2, 1;
From Philippe Deléham, Mar 25 2012: (Start)
(0, 2, -1/2, -1/2, 0, 0, ...) DELTA (1, 0, -1, 1, 0, 0, ...) begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  2,  1;
  0,  5,  4,  2,  1;
  0,  8,  8,  5,  2,  1;
  0, 13, 15, 11,  6,  2,  1;
  0, 21, 28, 23, 14,  7,  2,  1; (End)

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207610 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207611 *)
T[ n_, k_] := Which[k<0 || n<0, 0, n<2, Boole[k<=n] + Boole[k==0&&n==1], True, T[n, k] = T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* Michael Somos, Sep 19 2024 *)

Prog

(Python)
from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 28 2017
(PARI) {T(n, k) = if(k<0 || n<0, 0, n<2, (k<=n) + (k==0 && n==1), T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* Michael Somos, Sep 19 2024 */

Crossrefs

Cf. A207610, A208510.

Sequence in context: A322083 A058399 A209434 * A320973 A058400 A131344

Adjacent sequences: A207608 A207609 A207610 * A207612 A207613 A207614

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 19 2012

Status

approved