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A001874 Convolved Fibonacci numbers.
(Formerly M4174 N1738)
4

%I M4174 N1738

%S 1,6,27,98,315,924,2534,6588,16407,39430,91959,209034,464723,1013292,

%T 2171850,4584620,9546570,19635840,39940460,80421600,160437690,

%U 317354740,622844730,1213580820,2348773525,4517541378,8638447293,16428864606,31086197469,58539877020

%N Convolved Fibonacci numbers.

%C a(n) = (((-i)^n)/5!)*(d^5/dx^5)S(n+5,x)|_{x=i}, where i is the imaginary unit. Fifth derivative of Chebyshev S(n+5,x) polynomials evaluated at x=i multiplied by ((-i)^n)/5!. See A049310 for the S-polynomials. - _Wolfdieter Lang_, Apr 04 2007

%C a(n) is the number of weak compositions of n in which exactly 5 parts are 0 and all other parts are either 1 or 2. - _Milan Janjic_, Jun 28 2010

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001874/b001874.txt">Table of n, a(n) for n = 0..500</a>

%H P. J. Cameron, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,-10,30,6,-41,-6,30,10,-9,-6,-1).

%F G.f.: ( 1 - x - x^2 )^(-6).

%F a(n) = F'''''(n+5, 1)/5!, i.e., 1/5! times the 5th derivative of the (n+5)th Fibonacci polynomial evaluated at 1. - _T. D. Noe_, Jan 18 2006

%e G.f. = 1 + 6*x + 27*x^2 + 98*x^3 + 315*x^4 + 924*x^5 + 2534*x^6 + ...

%p a:= n-> (Matrix(12, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [6, -9, -10,

%p 30, 6, -41, -6, 30, 10, -9, -6, -1][i], 0)))^n)[1, 1]:

%p seq(a(n), n=0..31); # _Alois P. Heinz_, Aug 15 2008

%t nn = 30; t = CoefficientList[Series[1/(1 - x - x^2)^6, {x, 0, nn}], x] (* _T. D. Noe_, Aug 10 2012 *)

%o (Sage) taylor( mul(x/(1-x-x^2)^2 for i in range(1,4)),x,0,27) # _Zerinvary Lajos_, Jun 01 2009

%Y Cf. A049310.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_

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Last modified June 13 10:36 EDT 2021. Contains 344988 sequences. (Running on oeis4.)