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Expansion of (1-x)/(1 - 3*x + x^2)^2.
(Formerly M3886 N1595)
19

%I M3886 N1595 #108 Mar 01 2024 09:47:42

%S 1,5,19,65,210,654,1985,5911,17345,50305,144516,411900,1166209,

%T 3283145,9197455,25655489,71293590,197452746,545222465,1501460635,

%U 4124739581,11306252545,30928921224,84451726200,230204999425

%N Expansion of (1-x)/(1 - 3*x + x^2)^2.

%C a(n) = ((n+1)*F(2*n+3)+(2*n+3)*F(2*(n+1)))/5 with F(n)=A000045(n) (Fibonacci numbers). One half of odd-indexed A001629(n), n >= 2 (Fibonacci convolution).

%C Convolution of F(2n+1) (A001519) and F(2n+2) (A001906(n+1)). - _Graeme McRae_, Jun 07 2006

%C Number of reentrant corners along the lower contours of all directed column-convex polyominoes of area n+3 (a reentrant corner along the lower contour is a vertical step that is followed by a horizontal step). a(n) = Sum_{k=0..ceiling((n+1)/2)} k*A121466(n+3,k). - _Emeric Deutsch_, Aug 02 2006

%C From _Wolfdieter Lang_, Jan 02 2012: (Start)

%C a(n) = A024458(2*n), n >= 1 (bisection, even arguments).

%C a(n) is also the odd part of the bisection of the half-convolution of the sequence A000045(n+1), n >= 0, with itself. See a comment on A201204 for the definition of the half-convolution of a sequence with itself. There one also finds the rule for the o.g.f. which in this case is Chato(x)/2 with the o.g.f. Chato(x) = 2*(1-x)/(1-3*x+x^2)^2 of A001629(2*n+3), n >= 0.

%C (End)

%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 Vincenzo Librandi, <a href="/A001870/b001870.txt">Table of n, a(n) for n = 0..1000</a>

%H Elena Barcucci, Renzo Pinzani, and Renzo Sprugnoli , <a href="http://dx.doi.org/10.1007/3-540-56610-4_71">Directed column-convex polyominoes by recurrence relations</a>, Lecture Notes in Computer Science, No. 668, Springer, Berlin (1993), pp. 282-298.

%H Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/1803.06706">Descent distribution on Catalan words avoiding a pattern of length at most three</a>, arXiv:1803.06706 [math.CO], 2018.

%H Éva Czabarka, Rigoberto Flórez, and Leandro Junes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Florez/florez12.html">A Discrete Convolution on the Generalized Hosoya Triangle</a>, Journal of Integer Sequences, 18 (2015), #15.1.6.

%H Éva Czabarka, Rigoberto Flórez, Leandro Junes, and José L. Ramírez, <a href="https://doi.org/10.1016/j.disc.2018.06.032">Enumerations of peaks and valleys on non-decreasing Dyck paths</a>, Discrete Math. 341 (10) (2018), 2789-2807. See Cor. 6.

%H Emeric Deutsch and Helmut Prodinger, <a href="http://dx.doi.org/10.1016/S0304-3975(03)00222-6">A bijection between directed column-convex polyominoes and ordered trees of height at most three</a>, Theoretical Comp. Science, 307, 2003, 319-325.

%H Ricardo Gómez Aíza, <a href="https://arxiv.org/abs/2402.16111">Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis</a>, arXiv:2402.16111 [math.CO], 2024. See p. 23.

%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.

%H Pieter Moree, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Moree/moree12.htm">Convoluted Convolved Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2.

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H John Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>

%H John Riordan, <a href="/A002720/a002720_3.pdf">Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences</a>. Note that the sequences are identified by their N-numbers, not their A-numbers.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).

%F a(n) = Sum_{k=1..n+1} k*binomial(n+k+1, 2k). - _Emeric Deutsch_, Jun 11 2003

%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 10 2012

%F a(n) = (A238846(n) + A001871(n))/2. - _Philippe Deléham_, Mar 06 2014

%F a(n) = ((2*n-1)*Fibonacci(2*n) - n*Fibonacci(2*n-1))/5 [Czabarka et al.]. - _N. J. A. Sloane_, Sep 18 2018

%p A001870:=-(-1+z)/(z**2-3*z+1)**2; # _Simon Plouffe_ in his 1992 dissertation.

%t CoefficientList[Series[(1-x)/(1-3*x+x^2)^2,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 10 2012 *)

%t LinearRecurrence[{6,-11,6,-1},{1,5,19,65},30] (* _Harvey P. Dale_, Aug 17 2013 *)

%t With[{F=Fibonacci}, Table[((n+1)*F[2*n+3]+(2*n+3)*F[2*n+2])/5, {n,0,30}]] (* _G. C. Greubel_, Jul 15 2019 *)

%o (Magma) I:=[1, 5, 19, 65]; [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

%o (PARI) Vec((1-x)/(1-3*x+x^2)^2+O(x^30)) \\ _Charles R Greathouse IV_, Sep 23 2012

%o (Haskell)

%o a001870 n = a001870_list !! n

%o a001870_list = uncurry c $ splitAt 1 $ tail a000045_list where

%o c us vs'@(v:vs) = (sum $ zipWith (*) us vs') : c (v:us) vs

%o -- _Reinhard Zumkeller_, Oct 31 2013

%o (Sage) f=fibonacci; [((n+1)*f(2*n+3)+(2*n+3)*f(2*n+2))/5 for n in (0..30)] # _G. C. Greubel_, Jul 15 2019

%o (GAP) F:=Fibonacci;; List([0..30], n-> ((n+1)*F(2*n+3)+(2*n+3)*F(2*(n+1)))/5); # _G. C. Greubel_, Jul 15 2019

%Y a(n) = A060921(n+1, 1)/2.

%Y Partial sums of A030267. First differences of A001871.

%Y Cf. A121466.

%Y Cf. A023610.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Christian G. Bower_