A207605 Triangle of coefficients of polynomials u(n,x) jointly generated with A106195; see the Formula section.

1, 2, 4, 1, 8, 4, 1, 16, 12, 5, 1, 32, 32, 18, 6, 1, 64, 80, 56, 25, 7, 1, 128, 192, 160, 88, 33, 8, 1, 256, 448, 432, 280, 129, 42, 9, 1, 512, 1024, 1120, 832, 450, 180, 52, 10, 1, 1024, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 2048, 5120, 6912, 6400, 4424, 2364, 985, 316, 75, 12, 1

Offset: 1

Author

Clark Kimberling, Feb 19 2012

Keywords

• nonn
• tabf

Comments

Row sums: 1,2,5,13,... (odd-indexed Fibonacci numbers).

Alternating row sums: 1,2,3,5,... (Fibonacci numbers).

Subtriangle of the triangle given by (1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 22 2012

Examples

			First five rows:
   1
   2
   4   1
   8   4   1
  16  12   5   1
  32  32  18   6   1
First four polynomials u(n,x): 1, 2, 4 + x, 8 + 4x + x^2.
(1, 1, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:
   1
   1,  0
   2,  0,  0
   4,  1,  0,  0
   8,  4,  1,  0,  0
  16, 12,  5,  1,  0,  0
  32, 32, 18,  6,  1,  0,  0. - _Philippe Deléham_, Mar 22 2012
		

Links

• G. C. Greubel, Rows n = 1..102 of the triangle, flattened

Crossrefs

Cf. A001519, A106195.

Programs

Maple

CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):
T := (n,k) -> binomial(n, k)*hypergeom([-k,n-k], [-n], x):
P := [seq(add(simplify(T(n,k)),k=0..n), n=0..11)]:
seq(CoeffList(p), p in P); # Peter Luschny, Feb 16 2018

Mathematica

(* First program *)
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 + 1) v[n - 1, x]
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[%]  (* A207605 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A106195 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, 2^(n+1), If[k==n, 1, 2*T[n-1, k] + T[n-1, k-1] - T[n-2, k-1] ]]]; Join[{1}, Table[T[n, k], {n,0,10}, {k,0,n}]]//Flatten (* G. C. Greubel, Mar 15 2020 *)

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 + 1)*v(n - 1, x)
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

Sage

@CachedFunction
def T(n, k):
    if (k<0 or k>n): return 0
    elif k == 0: return 2^(n+1)
    elif k == n: return 1
    else: return 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1)
[1]+[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 15 2020

Formula

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

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(1,0) = 1, T(2,0) = 2, T(2,1) = 0. - Philippe Deléham, Mar 22 2012

G.f.: x*y*(1-x*y)/(1-x*y-2*x+x^2*y). - R. J. Mathar, Aug 11 2015

T(n,k) = [x^k] Sum_{k=0..n} binomial(n, k)*hypergeom([-k, n-k], [-n], x). - Peter Luschny, Feb 16 2018

Sum_{k=1..n} T(n,k) = Fibonacci(2*n-1), n >= 1, = (-1)^(n-1)*A099496(n-1). - G. C. Greubel, Mar 15 2020