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%I M3476 N1413 #85 Sep 08 2022 08:44:29
%S 1,4,14,40,105,256,594,1324,2860,6020,12402,25088,49963,98160,190570,
%T 366108,696787,1315072,2463300,4582600,8472280,15574520,28481220,
%U 51833600,93914325,169457708,304597382,545556512,973877245,1733053440,3075011478
%N Convolved Fibonacci numbers.
%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="/A001872/b001872.txt">Table of n, a(n) for n = 0..500</a>
%H D. Birmajer, J. Gil and M. Weiner, <a href="http://arxiv.org/abs/1405.7727">Linear recurrence sequences and their convolutions via Bell polynomials</a>, arXiv:1405.7727 [math.CO], 2014.
%H D. Birmajer, J. B. Gil, M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Gil/gil3.html">Linear Recurrence Sequences and Their Convolutions via Bell Polynomials</a>, Journal of Integer Sequences, 18 (2015), #15.1.2.
%H P. 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 V. E. Hoggatt, Jr. and M. Bicknell-Johnson, <a href="http://www.fq.math.ca/Scanned/15-2/hoggatt1.pdf">Fibonacci convolution sequences</a>, Fib. Quart., 15 (1977), 117-122.
%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences </a>, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
%H T. Mansour, <a href="http://arXiv.org/abs/math.CO/0301157">Generalization of some identities involving the Fibonacci numbers</a>, arXiv:math/0301157 [math.CO], 2003.
%H P. Moree, <a href="https://arxiv.org/abs/math/0311205">Convoluted convolved Fibonacci numbers</a>, arXiv:math/0311205 [math.CO], 2003.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-8,5,8,-2,-4,-1).
%F G.f.: 1/(1 - x - x^2)^4.
%F a(n) = A037027(n+3, 3) (Fibonacci convolution triangle).
%F a(n) = (n+5)*(n+3)*(4*(n+1)*F(n+2)+3*(n+2)*F(n+1))/150, F(n)=A000045(n). - _Wolfdieter Lang_, Apr 12 2000
%F For n > 3, a(n-3) = Sum_{h+i+j+k=n} F(h)*F(i)*F(j)*F(k). - _Benoit Cloitre_, Nov 01 2002
%F a(n) = F'''(n+3, 1)/6, i.e., 1/6 times the 3rd derivative of the (n+3)th Fibonacci polynomial evaluated at 1. - _T. D. Noe_, Jan 18 2006
%F a(n) = (((-i)^n)/3!)*(d^3/dx^3)S(n+3,x)|_{x=i}, where i is the imaginary unit. Third derivative of Chebyshev S(n+3,x) polynomial evaluated at x=i multiplied by ((-i)^(n-3))/3!. See A049310 for the S-polynomials. - _Wolfdieter Lang_, Apr 04 2007
%F a(n) = Sum_{i=ceiling(n/2)..n} (i+1)*(i+2)*(i+3)*binomial(i,n-i)/6. - _Vladimir Kruchinin_, Apr 26 2011
%F Recurrence: a(n) = 4*a(n-1) - 2*a(n-2) - 8*a(n-3) + 5*a(n-4) + 8*a(n-5) - 2*a(n-6) - 4*a(n-7) - a(n-8). - _Fung Lam_, May 11 2014
%F n*a(n) - (n+3)*a(n-1) - (n+6)*a(n-2) = 0, n > 1. - _Michael D. Weiner_, Nov 18 2014
%p a := n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [4,-2,-8,5, 8,-2,-4,-1][i] else 0 fi)^n)[1,1]; seq (a(n), n=0..29); # _Alois P. Heinz_, Aug 15 2008
%t CoefficientList[Series[1/(1 - x - x^2)^4, {x, 0, 100}], x] (* _Stefan Steinerberger_, Apr 15 2006 *)
%o (PARI) Vec( 1/(1 - x - x^2)^4 + O(x^66) ) \\ _Joerg Arndt_, May 12 2014
%o (Magma) [(n+5)*(n+3)*(4*(n+1)*Fibonacci(n+2)+3*(n+2)*Fibonacci(n+1))/150: n in [0..30]]; // _Vincenzo Librandi_, Nov 19 2014
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
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