This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A207605 #41 Apr 13 2020 04:10:14
%S A207605 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 A207605 33,8,1,256,448,432,280,129,42,9,1,512,1024,1120,832,450,180,52,10,1,
%U A207605 1024,2304,2816,2352,1452,681,242,63,11,1,2048,5120,6912,6400,4424,2364,985,316,75,12,1
%N A207605 Triangle of coefficients of polynomials u(n,x) jointly generated with A106195; see the Formula section.
%C A207605 Row sums: 1,2,5,13,... (odd-indexed Fibonacci numbers).
%C A207605 Alternating row sums: 1,2,3,5,... (Fibonacci numbers).
%C A207605 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 A207605 G. C. Greubel, <a href="/A207605/b207605.txt">Rows n = 1..102 of the triangle, flattened</a>
%F A207605 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 A207605 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 A207605 G.f.: x*y*(1-x*y)/(1-x*y-2*x+x^2*y). - _R. J. Mathar_, Aug 11 2015
%F A207605 T(n,k) = [x^k] Sum_{k=0..n} binomial(n, k)*hypergeom([-k, n-k], [-n], x). - _Peter Luschny_, Feb 16 2018
%F A207605 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 A207605 First five rows:
%e A207605 1
%e A207605 2
%e A207605 4 1
%e A207605 8 4 1
%e A207605 16 12 5 1
%e A207605 32 32 18 6 1
%e A207605 First four polynomials u(n,x): 1, 2, 4 + x, 8 + 4x + x^2.
%e A207605 (1, 1, 0, 0, 0, ...) DELTA (0, 0, 1, 0, 0, ...) begins:
%e A207605 1
%e A207605 1, 0
%e A207605 2, 0, 0
%e A207605 4, 1, 0, 0
%e A207605 8, 4, 1, 0, 0
%e A207605 16, 12, 5, 1, 0, 0
%e A207605 32, 32, 18, 6, 1, 0, 0. - _Philippe Deléham_, Mar 22 2012
%p A207605 CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):
%p A207605 T := (n,k) -> binomial(n, k)*hypergeom([-k,n-k], [-n], x):
%p A207605 P := [seq(add(simplify(T(n,k)),k=0..n), n=0..11)]:
%p A207605 seq(CoeffList(p), p in P); # _Peter Luschny_, Feb 16 2018
%t A207605 (* First program *)
%t A207605 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207605 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207605 v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x]
%t A207605 Table[Factor[u[n, x]], {n, 1, z}]
%t A207605 Table[Factor[v[n, x]], {n, 1, z}]
%t A207605 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207605 TableForm[cu]
%t A207605 Flatten[%] (* A207605 *)
%t A207605 Table[Expand[v[n, x]], {n, 1, z}]
%t A207605 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207605 TableForm[cv]
%t A207605 Flatten[%] (* A106195 *)
%t A207605 (* Second program *)
%t A207605 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 A207605 (Python)
%o A207605 from sympy import Poly
%o A207605 from sympy.abc import x
%o A207605 def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
%o A207605 def v(n, x): return 1 if n==1 else u(n - 1, x) + (x + 1)*v(n - 1, x)
%o A207605 def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
%o A207605 for n in range(1, 13): print(a(n)) # _Indranil Ghosh_, May 27 2017
%o A207605 (Sage)
%o A207605 @CachedFunction
%o A207605 def T(n, k):
%o A207605 if (k<0 or k>n): return 0
%o A207605 elif k == 0: return 2^(n+1)
%o A207605 elif k == n: return 1
%o A207605 else: return 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1)
%o A207605 [1]+[[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 15 2020
%Y A207605 Cf. A001519, A106195.
%K A207605 nonn,tabf
%O A207605 1,2
%A A207605 _Clark Kimberling_, Feb 19 2012