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%I #40 Sep 08 2022 08:45:13
%S 1,1,1,2,2,1,3,6,3,1,5,12,12,4,1,8,25,30,20,5,1,13,48,75,60,30,6,1,21,
%T 91,168,175,105,42,7,1,34,168,364,448,350,168,56,8,1,55,306,756,1092,
%U 1008,630,252,72,9,1,89,550,1530,2520,2730,2016,1050,360,90,10,1
%N Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k), 0 <= k <= n.
%C Triangle of coefficients of polynomials u(n,x) jointly generated with A209415; see the Formula section.
%C Column 1: Fibonacci numbers: F(n)=A000045(n)
%C Column 2: n*F(n)
%C Row sums: odd-indexed Fibonacci numbers
%C Alternating row sums: signed Fibonacci numbers
%C Coefficient of x^n in u(n,x): 1
%C Coefficient of x^(n-1) in u(n,x): n
%C Coefficient of x^(n-2) in u(n,x): n(n+1)
%C For a discussion and guide to related arrays, see A208510.
%C Subtriangle of the triangle given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 27 2012
%C Row n shows the coefficients of the numerator of the n-th derivative of (1/n!)*(x+1)/(1-x-x^2); see the Mathematica program. - _Clark Kimberling_, Oct 22 2019
%H G. C. Greubel, <a href="/A094441/b094441.txt">Rows n = 0..100 of triangle, flattened</a>
%H E. Kiliç, H. Belbachir, <a href="http://ekilic.etu.edu.tr/list/DoubleSums2.pdf">Generalized double binomial sums families by generating functions</a>, 2014.
%F Sum_{k=0..n} T(n,k)*x^k = A039834(n-1), A000045(n+1), A001519(n+1), A081567(n), A081568(n), A081569(n), A081570(n), A081571(n) for x = -1, 0, 1, 2, 3, 4, 5, 6 respectively. - _Philippe Deléham_, Dec 14 2009
%F From _Clark Kimberling_, Mar 09 2012: (Start)
%F A094441 shows the coefficient of the polynomials u(n,x) which are jointly generated with polynomials v(n,x) by these rules:
%F u(n,x) = x*u(n-1,x) + v(n-1,x),
%F v(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F (End)
%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1) - T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1 and T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Mar 27 2012
%F G.f. (1-x*y)/(1 - 2*x*y - x - x^2 + x^2*y + x^2*y^2). - _R. J. Mathar_, Aug 11 2015
%F From _G. C. Greubel_, Oct 30 2019: (Start)
%F T(n,k) = binomial(n,k)*Fibonacci(n-k+1).
%F Sum_{k=0..n} T(n,k) = Fibonacci(2*n+1).
%F Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * Fibonacci(n-1). (End)
%e First five rows:
%e 1;
%e 1, 1;
%e 2, 2, 1;
%e 3, 6, 3, 1;
%e 5, 12, 12, 4, 1;
%e First three polynomials v(n,x): 1, 1 + x, 2 + 2x + x^2.
%e From _Philippe Deléham_, Mar 27 2012: (Start)
%e (0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, ...) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 2, 1;
%e 0, 3, 6, 3, 1;
%e 0, 5, 12, 12, 4, 1. (End)
%p with(combinat); seq(seq(fibonacci(n-k+1)*binomial(n,k), k=0..n), n=0..12); # _G. C. Greubel_, Oct 30 2019
%t (* First program *)
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := 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[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A094441 *)
%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[%] (* A094442 *)
%t (* Next program outputs polynomials having coefficients T(n,k) *)
%t g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(x + 1)/(1 - x - x^2), {x, n}]]]
%t Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* _Clark Kimberling_, Oct 22 2019 *)
%t (* Second program *)
%t Table[Fibonacci[n-k+1]*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 30 2019 *)
%o (PARI) T(n,k) = binomial(n,k)*fibonacci(n-k+1);
%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Oct 30 2019
%o (Magma) [Binomial(n,k)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 30 2019
%o (Sage) [[binomial(n,k)*fibonacci(n-k+1) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Oct 30 2019
%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+1) ))); # _G. C. Greubel_, Oct 30 2019
%Y Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094442, A094443, A094444.
%K nonn,tabl
%O 0,4
%A _Clark Kimberling_, May 03 2004
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