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%I M1771 #29 Jun 01 2022 08:22:33
%S 1,2,7,26,107,468,2141,10124,49101,242934,1221427,6222838,32056215,
%T 166690696,873798681,4612654808,24499322137,130830894666,702037771647,
%U 3783431872018,20469182526595,111133368084892,605312629105205,3306633429423460,18111655081108453
%N Generalized Fibonacci numbers.
%C The Kn21 sums, see A180662, of the Schroeder triangle A033877 equal A006603(n) while the Kn3 sums equal A006603(2*n). The Kn22 sums, see A227504, and the Kn23 sums, see A227505, are also related to the sequence given above. - _Johannes W. Meijer_, Jul 15 2013
%C Typo on the right-hand side of Rogers's equation (1-x+x^2+x^3)*R^*(x) = R(x)+x: the sign in front of the x should be switched. - _R. J. Mathar_, Nov 23 2018
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H D. G. Rogers, <a href="https://dx.doi.org/10.1007/BFb0069192">A Schroeder triangle: three combinatorial problems</a>, in "Combinatorial Mathematics V (Melbourne 1976)", Lect. Notes Math. 622 (1976), pp. 175-196.
%F G.f.: (1 - x - 2x^2 - sqrt(1 - 6x + x^2))/(2x*(1 - x + x^2 + x^3)) = (A006318(x) - x)/(1 - x + x^2 + x^3).
%F a(n) = Sum_{k=1..floor(n/2)+1} k*(1/(n-k+2))*Sum_{i=0..n-2*k+2} C(n-k+2,i)*C(2*n-3*k-i+3,n-k+1). - _Vladimir Kruchinin_, Oct 23 2011
%F (n+1)*a(n) +(-7*n+2)*a(n-1) +4*(2*n-1)*a(n-2) +6*(-n+1)*a(n-3) +(-5*n+1)*a(n-4) +(n-2)*a(n-5)=0. - _R. J. Mathar_, Nov 23 2018
%p A006603 := n-> add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): seq(A006603(n), n=0..24); # _Johannes W. Meijer_, Jul 15 2013
%t CoefficientList[Series[(1-x-2x^2-Sqrt[1-6x+x^2])/(2x(1-x+x^2+x^3)),{x,0,30}],x] (* _Harvey P. Dale_, Jun 12 2016 *)
%o (Maxima) a(n):=sum((k*sum(binomial(n-k+2,i)*binomial(2*n-3*k-i+3,n-k+1),i,0,n-2*k+2))/(n-k+2),k,1,n/2+1); /* _Vladimir Kruchinin_, Oct 23 2011 */
%Y a(n) = abs(A080244(n-1)).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E More terms from _Emeric Deutsch_, Feb 28 2004
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