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 A093131 Binomial transform of Fibonacci(2n). 8
 0, 1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000, 40957275390625, 148184814453125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Second binomial transform of Fibonacci(n). - Paul Barry, Apr 22 2005 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..1791 S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014. M. Griffiths, Families of Sequences From a Class of Multinomial Sums, Journal of Integer Sequences, 15 (2012), #12.1.8. László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2. J. Pan, Multiple Binomial Transforms and Families of Integer Sequences , J. Int. Seq. 13 (2010), 10.4.2, F^(2) and absolute values of F^(-2). J. Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences , J. Int. Seq. 14 (2011) # 11.3.4, remark 14. Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019). Index entries for linear recurrences with constant coefficients, signature (5,-5). FORMULA G.f.: x/(1 - 5*x + 5*x^2). a(n) = (((5 + sqrt(5))/2)^n - ((5 - sqrt(5))/2)^n)/sqrt(5). a(n) = A093130(n)/2^n. a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*Fibonacci(j-k). - Paul Barry, Feb 15 2005 a(n) = Sum_{k=0..n} C(n, k)*2^k*Fibonacci(n-k) = Sum_{k=0..n} C(n, k)*2^(n-k) * Fibonacci(k). - Paul Barry, Apr 22 2005 a(n) = A030191(n-1), n > 0. - R. J. Mathar, Sep 05 2008 E.g.f.: 2*exp(5*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Ilya Gutkovskiy, Aug 11 2017 From Kai Wang, Dec 22 2019: (Start) a(n) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} 5^i*((i+j)!/(i!*j!)). a(n*k)/a(k) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} (-1)^(j*(k-1))*b(k)^i*((i+j)!/(i!*j!)). a((2*m+1)*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*k)*A020876((2*m-2*i)*k) + 5^(m*k). a(2*m*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*k)*A020876((2*m-2*i-1)*k}. a(m+r)*a(n+s) - a(m+s)*a(n+r) = -5^(n+s)*a(m-n)*a(r-s). a(m+r)*a(n+s) + a(m+s)*a(n+r) = (2*A020876(m+n+r+s) - 5^(n+s)*A020876(m-n)*A020876(r-s))/5. A020876(m+r)*A020876(n+s) - A020876(m+s)*A020876(n+r) = 5^(n+s+1)*a(m-n)*a(r-s). A020876(m+r)*A020876(n+s) - 5*a(m+s)*a(n+r) = 5^(n+s)*A020876(m-n)*A020876(r-s). A020876(m+r)*A020876(n+s) + 5*a(m+s)*a(n+r) = 2*A020876(m+n+r+s) + 5^(n+s+1)*a(m-n)*a(r-s). a(n)^2 - a(n+1)*a(n-1) = 5^(n-1). a(n)^2 - a(n+r)*a(n-r) = 5^(n-r)*a(r)^2. a(m)*a(n+1) - a(m+1)*a(n) = 5^n*a(m-n). a(m-n) = (a(m)*A020876(n) - A020876(m)*a(n))/(2*5^n). a(m+n) = (a(m)*A020876(n) + A020876(m)*a(n))/2. A020876(n)^2 - A020876(n+r)*A020876(n-r) = -5^(n-r+1)*a(r)^2. A020876(m)*A020876(n+1) - A020876(m+1)*A020876(n) = -5^(n+1)*a(m-n). A020876(m+n) - 5^(n)*A020876(m-n) = 5*a(m)*a(n). A020876(m-n) = (A020876(m)*A020876(n) - 5*a(m)*a(n))/(2*5^n). A020876(m+n) = (A020876(m)*A020876(n) + 5*a(m)*a(n))/2.  (End) a(2*n) = 5^n*Fibonacci(2*n), a(2*n+1) = 5^n*Lucas(2*n+1). - G. C. Greubel, Dec 27 2019 MAPLE seq(coeff(series(x/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 27 2019 MATHEMATICA CoefficientList[Series[x/(1-5x+5x^2), {x, 0, 30}], x] (* Michael De Vlieger, Dec 22 2019 *) Table[If[EvenQ[n], 5^(n/2)*Fibonacci[n], 5^((n-1)/2)*LucasL[n]], {n, 0, 30}] (* G. C. Greubel, Dec 27 2019 *) PROG (PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-5*x+5*x^2))) \\ G. C. Greubel, Dec 27 2019 (MAGMA) I:=[0, 1]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 27 2019 (Sage) def A093131_list(prec):     P. = PowerSeriesRing(ZZ, prec)     return P( x/(1-5*x+5*x^2) ).list() A093131_list(30) # G. C. Greubel, Dec 27 2019 (GAP) a:=[0, 1];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 27 2019 CROSSREFS Cf. A000032, A000045, A020876, A030191. Sequence in context: A092490 A094828 A030191 * A224422 A000344 A290922 Adjacent sequences:  A093128 A093129 A093130 * A093132 A093133 A093134 KEYWORD easy,nonn AUTHOR Paul Barry, Mar 23 2004 STATUS approved

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Last modified July 3 17:54 EDT 2022. Contains 355055 sequences. (Running on oeis4.)