A207605 - OEIS

%I #41 Apr 13 2020 04:10:14

%S 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,

%T 33,8,1,256,448,432,280,129,42,9,1,512,1024,1120,832,450,180,52,10,1,

%U 1024,2304,2816,2352,1452,681,242,63,11,1,2048,5120,6912,6400,4424,2364,985,316,75,12,1

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

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

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

%C 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

%H G. C. Greubel, <a href="/A207605/b207605.txt">Rows n = 1..102 of the triangle, flattened</a>

%F 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.

%F 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

%F G.f.: x*y*(1-x*y)/(1-x*y-2*x+x^2*y). - _R. J. Mathar_, Aug 11 2015

%F T(n,k) = [x^k] Sum_{k=0..n} binomial(n, k)*hypergeom([-k, n-k], [-n], x). - _Peter Luschny_, Feb 16 2018

%F 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

%e First five rows:

%e    1

%e    2

%e    4   1

%e    8   4   1

%e   16  12   5   1

%e   32  32  18   6   1

%e First four polynomials u(n,x): 1, 2, 4 + x, 8 + 4x + x^2.

%e (1, 1, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:

%e    1

%e    1,  0

%e    2,  0,  0

%e    4,  1,  0,  0

%e    8,  4,  1,  0,  0

%e   16, 12,  5,  1,  0,  0

%e   32, 32, 18,  6,  1,  0,  0. - _Philippe Deléham_, Mar 22 2012

%p CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):

%p T := (n,k) -> binomial(n, k)*hypergeom([-k,n-k], [-n], x):

%p P := [seq(add(simplify(T(n,k)),k=0..n), n=0..11)]:

%p seq(CoeffList(p), p in P); # _Peter Luschny_, Feb 16 2018

%t (* First program *)

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + v[n - 1, x]

%t v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x]

%t Table[Factor[u[n, x]], {n, 1, z}]

%t Table[Factor[v[n, x]], {n, 1, z}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%]  (* A207605 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%]  (* A106195 *)

%t (* Second program *)

%t 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 *)

%o (Python)

%o from sympy import Poly

%o from sympy.abc import x

%o def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)

%o def v(n, x): return 1 if n==1 else u(n - 1, x) + (x + 1)*v(n - 1, x)

%o def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]

%o for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 27 2017

%o (Sage)

%o @CachedFunction

%o def T(n, k):

%o     if (k<0 or k>n): return 0

%o     elif k == 0: return 2^(n+1)

%o     elif k == n: return 1

%o     else: return 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1)

%o [1]+[[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 15 2020

%Y Cf. A001519, A106195.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Feb 19 2012