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

1,2

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

Links

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

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

Example

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

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 *)

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 + 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

Crossrefs

Cf. A001519, A106195.

Sequence in context: A275486 A065278 A182896 * A112931 A121685 A125810

Adjacent sequences: A207602 A207603 A207604 * A207606 A207607 A207608

Keyword

nonn,tabf

Author

Clark Kimberling, Feb 19 2012

Status

approved