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 A001871 Expansion of 1/(1 - 3*x + x^2)^2. (Formerly M4166 N1733) 19
 1, 6, 25, 90, 300, 954, 2939, 8850, 26195, 76500, 221016, 632916, 1799125, 5082270, 14279725, 39935214, 111228804, 308681550, 853904015, 2355364650, 6480104231, 17786356776, 48715278000, 133167004200, 363372003625, 989900286774 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Convolution of A001906(n), n >= 1 (even-indexed Fibonacci numbers) with itself. A001787 and this sequence arise in counting ordered trees of height at most k where only the rightmost branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for A001787 and k = 4 for this sequence. Gives the number of 3412-avoiding permutations containing exactly one subsequence of type 321. - Dan Daly (ddaly(AT)du.edu), Apr 24 2008 REFERENCES Rigoberto Flórez, Leandro Junes, José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics (2019) Vol. 342, Issue 11, 3079-3097. See page 3092. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Rigoberto Flórez, Robinson Higuita, Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018. S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243, 2012. - From N. J. A. Sloane, May 09 2012 Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274) Eunjeong Lee, Mikiya Masuda, Seonjeong Park, On Schubert varieties of complexity one, arXiv:2009.02125 [math.AT], 2020. Valentin Ovsienko, Serge Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, arXiv:1705.01623 [math.CO], 2017. See p. 9. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. J. Riordan, Notes to N. J. A. Sloane, Jul. 1968 John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers. Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1). FORMULA a(n) = (2*(2*n+1)*F(2*(n+1))+3*(n+1)*F(2*n+1))/5 with F(n) = A000045 (Fibonacci numbers). a(n) = -a(-4-n) = ((4n+2)F(2n) + (7n+5)F(2n+1))/5 with F(n) = A000045 (Fibonacci numbers). a(n) = (2*a(n-1) + (n+1)*F(2n+4))/3, where F(n) = A000045 (Fibonacci numbers). - Emeric Deutsch, Oct 08 2002 G.f.: 1/(1 - 3*x + x^2)^2. a(n) = (Sum_{k=0..n} S(k, 3)S(n-k, 3)) S(n, x) = U(n, x/2) Chebyshev polynomials of 2nd kind, A049310. - Paul Barry, Nov 14 2003 a(n) = Sum_{k=1..n+1} F(2k)*F(2(n-k+2)) where F(k) is the k-th Fibonacci number. - Dan Daly (ddaly(AT)du.edu), Apr 24 2008 a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - Vincenzo Librandi, Mar 14 2011 a(n) = 2*A001870(n) - A238846(n). - Philippe Deléham, Mar 06 2014 MAPLE A001871:=1/(z**2-3*z+1)**2; # Simon Plouffe in his 1992 dissertation f:= gfun:-rectoproc({a(n)=6*a(n-1)-11*a(n-2)+6*a(n-3)-a(n-4), a(0)=1, a(1)=6, a(2)=25, a(3)=90}, a(n), remember): map(f, [\$0..50]); # Robert Israel, May 05 2017 MATHEMATICA CoefficientList[Series[1/(1-3x+x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 10 2012 *) PROG (PARI) a(n)=((4*n+2)*fibonacci(2*n)+(7*n+5)*fibonacci(2*n+1))/5 (MAGMA) I:=[1, 6, 25, 90]; [n le 4 select I[n] else 6*Self(n-1)-11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 10 2012 (PARI) Vec(1/(1-3*x+x^2)^2 + O(x^100)) \\ Altug Alkan, Oct 31 2015 CROSSREFS Partial sums of A001870 (one half of odd-indexed A001629(n), n >= 2, Fibonacci convolution). Cf. A001629. Sequence in context: A056279 A055337 A309946 * A000392 A099948 A333017 Adjacent sequences:  A001868 A001869 A001870 * A001872 A001873 A001874 KEYWORD nonn,easy,changed AUTHOR EXTENSIONS Additional comments from Wolfdieter Lang, Apr 07 2000 STATUS approved

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Last modified December 5 16:22 EST 2020. Contains 338954 sequences. (Running on oeis4.)