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A005807 Sum of adjacent Catalan numbers.

%I A005807 M0850 #76 Jun 21 2025 16:11:46
%S A005807 2,3,7,19,56,174,561,1859,6292,21658,75582,266798,950912,3417340,
%T A005807 12369285,45052515,165002460,607283490,2244901890,8331383610,
%U A005807 31030387440,115948830660,434542177290,1632963760974,6151850548776
%N A005807 Sum of adjacent Catalan numbers.
%C A005807 The aerated sequence has Hankel transform F(n+2)*F(n+3) (A001654(n+2)). - _Paul Barry_, Nov 04 2008
%D A005807 D. E. Knuth, personal communication.
%D A005807 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005807 Vincenzo Librandi, <a href="/A005807/b005807.txt">Table of n, a(n) for n = 0..200</a>
%H A005807 Daniel Birmajer, Juan B. Gil, Michael D. Weiner, <a href="http://arxiv.org/abs/1606.02183">On rational Dyck paths and the enumeration of factor-free Dyck words</a>, arXiv:1606.02183 [math.CO], (7-June-2016); see p. 9
%H A005807 Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Ivkovic/ivkovic3.html">Catalan Numbers, the Hankel Transform and Fibonacci Numbers</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3
%H A005807 Sergio Falcon, <a href="http://www.mathnet.or.kr/mathnet/thesis_file/CKMS-28-4-827-832.pdf">Catalan transform of the K-Fibonacci sequence</a>, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
%H A005807 Manuel Flores, Yuta Kimura, Baptiste Rognerud, <a href="https://arxiv.org/abs/2004.04726">Combinatorics of quasi-hereditary structures</a>, arXiv:2004.04726 [math.RT], 2020.
%H A005807 Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>
%H A005807 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]
%H A005807 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=431">Encyclopedia of Combinatorial Structures 431</a>
%H A005807 Qi Wang, <a href="https://arxiv.org/abs/1910.01937">Tau-tilting finite simply connected algebras</a>, arXiv:1910.01937 [math.RT], 2019.
%H A005807 Fumitaka Yura, <a href="http://arxiv.org/abs/1411.6972">Hankel Determinant Solution for Elliptic Sequence</a>, arXiv:1411.6972 [nlin.SI], (25-November-2014); see p. 7.
%F A005807 a(n) = C(n)+C(n+1) = ((5*n+4)*(2*n)!)/(n!*(n+2)!).
%F A005807 G.f. A(x) satisfies x^2*A(x)^2 + (x-1)*A(x) + (x+2) = 0. - _Michael Somos_, Sep 11 2003
%F A005807 G.f.: (1-x - (1+x)*sqrt(1-4*x)) / (2*x^2) = (4+2*x) / (1-x + (1+x)*sqrt(1-4*x)). a(n)*(n+2)*(5*n-1) = a(n-1)*2*(2*n-1)*(5*n+4), n>0. - _Michael Somos_, Sep 11 2003
%F A005807 a(n) ~ 5*Pi^(-1/2)*n^(-3/2)*2^(2*n)*{1 - 93/40*n^-1 + 625/128*n^-2 - 10227/1024*n^-3 + 661899/32768*n^-4 ...}. - Joe Keane (jgk(AT)jgk.org), Sep 13 2002
%F A005807 G.f.: c(x)*(1+c(x))= (-1 +(1+x)*c(x))/x with the g.f. c(x) of A000108 (Catalan).
%F A005807 a(n) = binomial(2*n,n)/(n+1)*hypergeom([-1,n+1/2],[n+2],-4). - _Peter Luschny_, Aug 15 2012
%F A005807 D-finite with recurrence (n+2)*a(n) + (-3*n-2)*a(n-1) + 2*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Dec 02 2012
%F A005807 0 = a(n)*(+16*a(n+1) + 38*a(n+2) - 18*a(n+3)) + a(n+1)*(-14*a(n+1) + 19*a(n+2) - 7*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3)) for all n>=0. - _Michael Somos_, Jan 17 2015
%F A005807 0 = a(n)^2*(+368*a(n+1) - 182*a(n+2)) + a(n)*a(n+1)*(-306*a(n+1) + 317*a(n+2)) + a(n)*a(n+2)*(-77*a(n+2)) + a(n+1)^2*(-14*a(n+1) - 6*a(n+2)) + a(n+1)*a(n+2)*(+8*a(n+2)) for all n>=0. - _Michael Somos_, Jan 17 2015
%F A005807 E.g.f.: (BesselI(0,2*x) - (x - 1)*BesselI(1,2*x)/x)*exp(2*x). - _Ilya Gutkovskiy_, Jun 08 2016
%F A005807 G.f. with 1 prepended: Let E(x) = exp( Sum_{n >= 1} binomial(5*n,2*n)*x^n/n ). Then A(x) = ( x/series reversion of x*E(x) )^(1/5) = ( x/series reversion of x*D(x)^5 )^(1/5), where D(x) = 1 + 2*x + 23*x^2 + 371*x^3 + ... is the o.g.f. for A060941 .... Cf. A274052 and A274244. - _Peter Bala_, Jan 01 2020
%e A005807 G.f. = 2 + 3*x+ 7*x^2 + 19*x^3 + 56*x^4 + 174*x^5 + 561*x^6 + 1859*x^7 + ...
%p A005807 A005807List := proc(m) local A, P, n; A := [2,3]; P := [2,3];
%p A005807 for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-2]]);
%p A005807 A := [op(A), P[-1]] od; A end: A005807List(25); # _Peter Luschny_, Mar 26 2022
%t A005807 a[n_]:=Binomial[2*n, n]*(5*n+4)/(n+1)/(n+2); (* _Vladimir Joseph Stephan Orlovsky_, Dec 13 2008 *)
%t A005807 a[ n_] := If[ n < 0, 0, CatalanNumber[n] + CatalanNumber[n + 1]]; (* _Michael Somos_, Jan 17 2015 *)
%t A005807 Total/@Partition[CatalanNumber[Range[0,30]],2,1] (* _Harvey P. Dale_, Jun 21 2025 *)
%o A005807 (PARI) {a(n) = if( n<0, 0, binomial(2*n, n) * (5*n+4) / ((n+1) * (n+2)))};
%o A005807 (Sage) [catalan_number(i)+catalan_number(i+1) for i in range(0,25)] # _Zerinvary Lajos_, May 17 2009
%o A005807 (Magma) [((5*n+4)*Factorial(2*n))/(Factorial(n)*Factorial(n+2)): n in [0..30] ];  // _Vincenzo Librandi_, Aug 19 2011
%o A005807 (Python)
%o A005807 from __future__ import division
%o A005807 A005807_list, b = [], 2
%o A005807 for n in range(10**3):
%o A005807     A005807_list.append(b)
%o A005807     b = b*(4*n+2)*(5*n+9)//((n+3)*(5*n+4)) # _Chai Wah Wu_, Jan 28 2016
%Y A005807 Cf. A000108.
%Y A005807 Cf. A071716, A000778, A060941, A274052, A274244.
%K A005807 nonn
%O A005807 0,1
%A A005807 _N. J. A. Sloane_
%E A005807 More terms from Joe Keane (jgk(AT)jgk.org), Feb 08 2000
%E A005807 Asymptotic series corrected and extended by _Michael Somos_, Sep 11 2003