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A001870 Expansion of (1-x)/(1 - 3*x + x^2)^2.

%I A001870 M3886 N1595 #118 Mar 05 2025 02:02:16
%S A001870 1,5,19,65,210,654,1985,5911,17345,50305,144516,411900,1166209,
%T A001870 3283145,9197455,25655489,71293590,197452746,545222465,1501460635,
%U A001870 4124739581,11306252545,30928921224,84451726200,230204999425
%N A001870 Expansion of (1-x)/(1 - 3*x + x^2)^2.
%C A001870 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 A001870 Convolution of F(2n+1) (A001519) and F(2n+2) (A001906(n+1)). - _Graeme McRae_, Jun 07 2006
%C A001870 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 A001870 From _Wolfdieter Lang_, Jan 02 2012: (Start)
%C A001870 a(n) = A024458(2*n), n >= 1 (bisection, even arguments).
%C A001870 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 A001870 (End)
%D A001870 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001870 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001870 Vincenzo Librandi, <a href="/A001870/b001870.txt">Table of n, a(n) for n = 0..1000</a>
%H A001870 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 A001870 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 A001870 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 A001870 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 A001870 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 A001870 Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Florez/florez51.html">Counting Subwords in Non-Decreasing Dyck Paths</a>, Journal of Integer Sequences, Vol. 28 (2025), Article 25.1.6. See pp. 5-6, 14-15, 17, 19.
%H A001870 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 A001870 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 A001870 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 A001870 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 A001870 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A001870 John Riordan, <a href="/A001820/a001820.pdf">Notes to N. J. A. Sloane, Jul. 1968</a>
%H A001870 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 A001870 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F A001870 a(n) = Sum_{k=1..n+1} k*binomial(n+k+1, 2k). - _Emeric Deutsch_, Jun 11 2003
%F A001870 a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 10 2012
%F A001870 a(n) = (A238846(n) + A001871(n))/2. - _Philippe Deléham_, Mar 06 2014
%F A001870 a(n) = ((2*n-1)*Fibonacci(2*n) - n*Fibonacci(2*n-1))/5 [Czabarka et al.]. - _N. J. A. Sloane_, Sep 18 2018
%F A001870 E.g.f.: exp(3*x/2)*(5*(5 + 11*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(13 + 25*x)*sinh(sqrt(5)*x/2))/25. - _Stefano Spezia_, Mar 04 2025
%p A001870 A001870:=-(-1+z)/(z**2-3*z+1)**2; # _Simon Plouffe_ in his 1992 dissertation.
%t A001870 CoefficientList[Series[(1-x)/(1-3*x+x^2)^2,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 10 2012 *)
%t A001870 LinearRecurrence[{6,-11,6,-1},{1,5,19,65},30] (* _Harvey P. Dale_, Aug 17 2013 *)
%t A001870 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 A001870 (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 A001870 (PARI) Vec((1-x)/(1-3*x+x^2)^2+O(x^30)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A001870 (Haskell)
%o A001870 a001870 n = a001870_list !! n
%o A001870 a001870_list = uncurry c $ splitAt 1 $ tail a000045_list where
%o A001870    c us vs'@(v:vs) = (sum $ zipWith (*) us vs') : c (v:us) vs
%o A001870 -- _Reinhard Zumkeller_, Oct 31 2013
%o A001870 (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 A001870 (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 A001870 a(n) = A060921(n+1, 1)/2.
%Y A001870 Partial sums of A030267. First differences of A001871.
%Y A001870 Cf. A121466.
%Y A001870 Cf. A023610.
%K A001870 nonn,easy
%O A001870 0,2
%A A001870 _N. J. A. Sloane_
%E A001870 More terms from _Christian G. Bower_