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%I M4412 N1865 #52 Aug 05 2023 16:53:28
%S 1,7,35,140,490,1554,4578,12720,33705,85855,211519,506408,1182650,
%T 2702350,6056850,13343820,28947240,61926900,130814600,273163100,
%U 564415390,1154933230,2342193350,4710707400,9401674275,18629923053,36670044621,71728832280,139485074370
%N Convolved Fibonacci numbers.
%C a(n) = (((-i)^n)/6!)*((d^5/dx^5) S(n+6,x))|_{x=i}. Sixth derivative of Chebyshev S(n+6,x) polynomials evaluated at x=i (imaginary unit) multiplied by ((-i)^n)/6!. See A049310 for the S-polynomials. - _Wolfdieter Lang_, Apr 04 2007
%C a(n) = number of weak compositions of n in which exactly 6 part 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).
%D P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.
%H T. D. Noe, <a href="/A001875/b001875.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 P. R. Stein and M. S. Waterman, <a href="/A001006/a001006_4.pdf">On some new sequences generalizing the Catalan and Motzkin numbers</a> [Corrected annotated scanned copy]
%H M. S. Waterman, <a href="http://www.cmb.usc.edu/people/msw/Waterman.html">Home Page</a> (contains copies of his papers)
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (7, -14, -7, 49, -14, -77, 29, 77, -14, -49, -7, 14, 7, 1).
%F G.f.: (1 - x - x^2)^(-7).
%F a(n) = F''''''(n+6, 1)/6!, i.e., 1/6! times the 6th derivative of the (n+6)th Fibonacci polynomial evaluated at 1. - _T. D. Noe_, Jan 18 2006
%F a(n) = Sum_{k=ceiling(n/2)..n} (k+1)*(k+2)*(k+3)*(k+4)*(k+5)*(k+6)*binomial(k,n-k)/720. - _Vladimir Kruchinin_, Apr 26 2011
%p a:= n-> (Matrix(14, (i,j)-> if (i=j-1) then 1 elif j=1 then [7,-14,-7,49, -14,-77,29,77, -14,-49,-7,14, 7,1][i] else 0 fi)^n)[1,1]: seq (a(n), n=0..22); # _Alois P. Heinz_, Aug 15 2008
%t nn = 30; CoefficientList[Series[1/(1 - x - x^2)^7, {x, 0, nn}], x] (* _T. D. Noe_, Aug 10 2012 *)
%t LinearRecurrence[{7,-14,-7,49,-14,-77,29,77,-14,-49,-7,14,7,1},{1,7,35,140,490,1554,4578,12720,33705,85855,211519,506408,1182650,2702350},30] (* _Harvey P. Dale_, Aug 05 2023 *)
%K nonn
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
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