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A207605
Triangle of coefficients of polynomials u(n,x) jointly generated with A106195; see the Formula section.
3
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
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
Sequence in context: A275486 A065278 A182896 * A112931 A121685 A125810
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Feb 19 2012
STATUS
approved