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 A094441 Triangular array T(n,k) = Fibonacci(n+1-k)*C(n,k), 0 <= k <= n. 17
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Triangle of coefficients of polynomials u(n,x) jointly generated with A209415; see the Formula section. Column 1: Fibonacci numbers: F(n)=A000045(n) Column 2: n*F(n) Row sums: odd-indexed Fibonacci numbers Alternating row sums: signed Fibonacci numbers Coefficient of x^n in u(n,x): 1 Coefficient of x^(n-1) in u(n,x): n Coefficient of x^(n-2) in u(n,x): n(n+1) For a discussion and guide to related arrays, see A208510. 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 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 LINKS G. C. Greubel, Rows n = 0..100 of triangle, flattened E. Kiliç, H. Belbachir, Generalized double binomial sums families by generating functions, 2014. 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) EXAMPLE 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) MAPLE with(combinat); seq(seq(fibonacci(n-k+1)*binomial(n, k), k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019 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 *) PROG (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 (Magma) [Binomial(n, k)*Fibonacci(n-k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019 (Sage) [[binomial(n, k)*fibonacci(n-k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, k)*Fibonacci(n-k+1) ))); # G. C. Greubel, Oct 30 2019 CROSSREFS Cf. A000045, A094435, A094436, A094437, A094438, A094439, A094440, A094442, A094443, A094444. Sequence in context: A134399 A094436 A286012 * A107230 A159830 A293472 Adjacent sequences: A094438 A094439 A094440 * A094442 A094443 A094444 KEYWORD nonn,tabl AUTHOR Clark Kimberling, May 03 2004 STATUS approved

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Last modified December 1 23:26 EST 2023. Contains 367503 sequences. (Running on oeis4.)