A094441 Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k), 0 <= k <= n.
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, 91, 168, 175, 105, 42, 7, 1, 34, 168, 364, 448, 350, 168, 56, 8, 1, 55, 306, 756, 1092, 1008, 630, 252, 72, 9, 1, 89, 550, 1530, 2520, 2730, 2016, 1050, 360, 90, 10, 1
Offset: 0
Examples
First five rows: 1; 1, 1; 2, 2, 1; 3, 6, 3, 1; 5, 12, 12, 4, 1; First three polynomials v(n,x): 1, 1 + x, 2 + 2x + x^2. From _Philippe Deléham_, Mar 27 2012: (Start) (0, 1, 1, -1, 0, 0, 0, ...) DELTA (1, 0, 0, 1, 0, 0, 0, ...) begins: 1; 0, 1; 0, 1, 1; 0, 2, 2, 1; 0, 3, 6, 3, 1; 0, 5, 12, 12, 4, 1. (End)
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
- E. Kiliç, H. Belbachir, Generalized double binomial sums families by generating functions, 2014.
Crossrefs
Programs
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GAP
Flat(List([0..12], n-> List([0..n], k-> Binomial(n,k)*Fibonacci(n-k+1) ))); # G. C. Greubel, Oct 30 2019
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Magma
[Binomial(n,k)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019
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Maple
with(combinat); seq(seq(fibonacci(n-k+1)*binomial(n,k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019
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Mathematica
(* First program *) u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := x*u[n - 1, x] + v[n - 1, x]; v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%] (* A094441 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A094442 *) (* Next program outputs polynomials having coefficients T(n,k) *) g[x_, n_] := Numerator[(-1)^(n + 1) Factor[D[(x + 1)/(1 - x - x^2), {x, n}]]] Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* Clark Kimberling, Oct 22 2019 *) (* Second program *) Table[Fibonacci[n-k+1]*Binomial[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 30 2019 *) -
PARI
T(n,k) = binomial(n,k)*fibonacci(n-k+1); for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Oct 30 2019
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Sage
[[binomial(n,k)*fibonacci(n-k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019
Formula
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
From Clark Kimberling, Mar 09 2012: (Start)
A094441 shows the coefficient of the polynomials u(n,x) which are jointly generated with polynomials v(n,x) by these rules:
u(n,x) = 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.
(End)
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
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
From G. C. Greubel, Oct 30 2019: (Start)
T(n,k) = binomial(n,k)*Fibonacci(n-k+1).
Sum_{k=0..n} T(n,k) = Fibonacci(2*n+1).
Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * Fibonacci(n-1). (End)
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