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 A006603 Generalized Fibonacci numbers. (Formerly M1771) 7
 1, 2, 7, 26, 107, 468, 2141, 10124, 49101, 242934, 1221427, 6222838, 32056215, 166690696, 873798681, 4612654808, 24499322137, 130830894666, 702037771647, 3783431872018, 20469182526595, 111133368084892, 605312629105205, 3306633429423460, 18111655081108453 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 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 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Table of n, a(n) for n=0..24. D. G. Rogers, A Schroeder triangle: three combinatorial problems, in "Combinatorial Mathematics V (Melbourne 1976)", Lect. Notes Math. 622 (1976), pp. 175-196. FORMULA 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). 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 (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 MAPLE 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 MATHEMATICA 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 *) PROG (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 */ CROSSREFS a(n) = abs(A080244(n-1)). Sequence in context: A000151 A150568 A102319 * A080244 A124542 A003447 Adjacent sequences: A006600 A006601 A006602 * A006604 A006605 A006606 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Simon Plouffe EXTENSIONS More terms from Emeric Deutsch, Feb 28 2004 STATUS approved

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Last modified September 23 04:30 EDT 2023. Contains 365532 sequences. (Running on oeis4.)