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A001873 Convolved Fibonacci numbers.

%I A001873 M3899 N1600 #63 Jan 05 2025 19:51:32
%S A001873 1,5,20,65,190,511,1295,3130,7285,16435,36122,77645,163730,339535,
%T A001873 693835,1399478,2790100,5504650,10758050,20845300,40075630,76495450,
%U A001873 145052300,273381350,512347975,955187033,1772132390,3272875935,6018885570,11024814945,20118711993
%N A001873 Convolved Fibonacci numbers.
%C A001873 a(n) = (((-i)^n)/4!)*(d^4/dx^4)S(n+4,x)|_{x=i}, where i is the imaginary unit. Fourth derivative of Chebyshev S(n+4,x) polynomial evaluated at x=i multiplied by ((-i)^n)/4!. See A049310 for the S-polynomials. - _Wolfdieter Lang_, Apr 04 2007
%C A001873 a(n) = number of weak compositions of n in which exactly 4 parts are 0 and all other parts are either 1 or 2. - _Milan Janjic_, Jun 28 2010
%D A001873 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 101.
%D A001873 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001873 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001873 T. D. Noe, <a href="/A001873/b001873.txt">Table of n, a(n) for n = 0..500</a>
%H A001873 Peter J. Cameron, <a href="http://www.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 A001873 Verner E. Hoggatt, Jr. and Marjorie Bicknell-Johnson, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-2/hoggatt1.pdf">Fibonacci convolution sequences</a>, Fib. Quart., 15 (1977), 117-122.
%H A001873 Pieter Moree, <a href="https://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.
%H A001873 Mihai Prunescu and Lorenzo Sauras-Altuzarra, <a href="https://arxiv.org/abs/2405.04083">On the representation of C-recursive integer sequences by arithmetic terms</a>, arXiv:2405.04083 [math.LO], 2024. See p. 18.
%H A001873 Paul R. Stein and Michael S. Waterman, <a href="http://dx.doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1979), 261-272.
%H A001873 Paul R. Stein and Michael S. Waterman, <a href="/A001006/a001006_4.pdf">On some new sequences generalizing the Catalan and Motzkin numbers</a> [Corrected annotated scanned copy]
%H A001873 Michael S. Waterman, <a href="http://www.cmb.usc.edu/people/msw/Waterman.html">Home Page</a> (contains copies of his papers)
%H A001873 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,-10,15,11,-15,-10,5,5,1).
%F A001873 G.f.: 1/(1-x-x^2)^5.
%F A001873 From _Wolfdieter Lang_, Nov 29 2002: (Start)
%F A001873 a(n) = Sum_{m=0.. floor(n/2)} binomial(4+n-m, 4)*binomial(n-m, m).
%F A001873 a(n) = ((1368 + 970*n + 215*n^2 + 15*n^3)*(n+1)*F(n+2) + 2*(408 + 305*n + 70*n^2 + 5*n^3)*(n+2)*F(n+1))/(4!*5^3), with F(n) = A000045(n). (End)
%F A001873 a(n) = F''''(n+4, 1)/24, i.e., 1/24 times the 4th derivative of the (n+4)th Fibonacci polynomial evaluated at 1. - _T. D. Noe_, Jan 18 2006
%F A001873 Recurrence: a(n) = 5*a(n-1) - 5*a(n-2) - 10*a(n-3) + 15*a(n-4) + 11*a(n-5) - 15*a(n-6) - 10*a(n-7) + 5*a(n-8) + 5*a(n-9) + a(n-10). - _Fung Lam_, May 11 2014
%F A001873 For n > 1, a(n) = (4/n+1)*a(n-1)+(8/n+1)*a(n-2). - _Tani Akinari_, Sep 14 2023
%p A001873 a:= n-> (Matrix(10, (i, j)-> `if`(i=j-1, 1, `if`(j=1, [5, -5, -10, 15, 11, -15, -10, 5, 5, 1][i], 0 )))^n)[1, 1]: seq(a(n), n=0..40); # _Alois P. Heinz_, Aug 15 2008
%t A001873 nn = 30; CoefficientList[Series[1/(1 - x - x^2)^5, {x, 0, nn}], x] (* _T. D. Noe_, Aug 10 2012 *)
%t A001873 LinearRecurrence[{5,-5,-10,15,11,-15,-10,5,5,1},{1,5,20,65,190,511,1295,3130,7285,16435},40] (* _Harvey P. Dale_, Aug 10 2021 *)
%o A001873 (Maxima) a[n]:=if n<2 then 4*n+1 else (4/n+1)*a[n-1]+(8/n+1)*a[n-2];
%o A001873 makelist(a[n],n,0,50); /* _Tani Akinari_, Sep 14 2023 */
%K A001873 nonn
%O A001873 0,2
%A A001873 _N. J. A. Sloane_
%E A001873 More terms from _Wolfdieter Lang_, Nov 29 2002