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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 19 02:22 EDT 2020. Contains 337175 sequences. (Running on oeis4.)