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A005807
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Sum of adjacent Catalan numbers.
(Formerly M0850)
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20
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2, 3, 7, 19, 56, 174, 561, 1859, 6292, 21658, 75582, 266798, 950912, 3417340, 12369285, 45052515, 165002460, 607283490, 2244901890, 8331383610, 31030387440, 115948830660, 434542177290, 1632963760974, 6151850548776
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OFFSET
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0,1
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COMMENTS
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The aerated sequence has Hankel transform F(n+2)*F(n+3) (A001654(n+2)). - Paul Barry, Nov 04 2008
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REFERENCES
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D. E. Knuth, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, On rational Dyck paths and the enumeration of factor-free Dyck words, arXiv:1606.02183 [math.CO], (7-June-2016); see p. 9
Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, Catalan Numbers, the Hankel Transform and Fibonacci Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3
Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.
Manuel Flores, Yuta Kimura, Baptiste Rognerud, Combinatorics of quasi-hereditary structures, arXiv:2004.04726 [math.RT], 2020.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 431
Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019.
Fumitaka Yura, Hankel Determinant Solution for Elliptic Sequence, arXiv:1411.6972 [nlin.SI], (25-November-2014); see p. 7.
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FORMULA
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a(n) = C(n)+C(n+1) = ((5*n+4)*(2*n)!)/(n!*(n+2)!).
G.f. A(x) satisfies x^2*A(x)^2 + (x-1)*A(x) + (x+2) = 0. - Michael Somos, Sep 11 2003
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
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
G.f.: c(x)*(1+c(x))= (-1 +(1+x)*c(x))/x with the g.f. c(x) of A000108 (Catalan).
a(n) = binomial(2*n,n)/(n+1)*hypergeom([-1,n+1/2],[n+2],-4). - Peter Luschny, Aug 15 2012
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
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
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
E.g.f.: (BesselI(0,2*x) - (x - 1)*BesselI(1,2*x)/x)*exp(2*x). - Ilya Gutkovskiy, Jun 08 2016
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
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EXAMPLE
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G.f. = 2 + 3*x+ 7*x^2 + 19*x^3 + 56*x^4 + 174*x^5 + 561*x^6 + 1859*x^7 + ...
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MATHEMATICA
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a[n_]:=Binomial[2*n, n]*(5*n+4)/(n+1)/(n+2); (* Vladimir Joseph Stephan Orlovsky, Dec 13 2008 *)
a[ n_] := If[ n < 0, 0, CatalanNumber[n] + CatalanNumber[n + 1]]; (* Michael Somos, Jan 17 2015 *)
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PROG
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(PARI) {a(n) = if( n<0, 0, binomial(2*n, n) * (5*n+4) / ((n+1) * (n+2)))};
(Sage) [catalan_number(i)+catalan_number(i+1) for i in range(0, 25)] # Zerinvary Lajos, May 17 2009
(MAGMA) [((5*n+4)*Factorial(2*n))/(Factorial(n)*Factorial(n+2)): n in [0..30] ]; // Vincenzo Librandi, Aug 19 2011
(Python)
from __future__ import division
A005807_list, b = [], 2
for n in range(10**3):
A005807_list.append(b)
b = b*(4*n+2)*(5*n+9)//((n+3)*(5*n+4)) # Chai Wah Wu, Jan 28 2016
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CROSSREFS
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Cf. A000108.
Cf. A071716, A000778, A060941, A274052, A274244.
Sequence in context: A033844 A037028 A052919 * A167422 A060276 A337187
Adjacent sequences: A005804 A005805 A005806 * A005808 A005809 A005810
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Joe Keane (jgk(AT)jgk.org), Feb 08 2000
Asymptotic series corrected and extended by Michael Somos, Sep 11 2003
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STATUS
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approved
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