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A005807
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Sum of adjacent Catalan numbers.
(Formerly M0850)
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10
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| The aerated sequence has Hankel transform F(n+2)*F(n+3) (A001654(n+2)). [From Paul Barry, Nov 04 2008]
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REFERENCES
| 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
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
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
Guo-Niu Han, Enumeration of Standard Puzzles
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 431
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FORMULA
| 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).
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MATHEMATICA
| a[n_]:=Binomial[2*n, n]*(5*n+4)/(n+1)/(n+2); [From Vladimir Orlovsky, Dec 13 2008]
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PROG
| (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 xrange(0, 25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 2009]
(MAGMA) [((5*n+4)*Factorial(2*n))/(Factorial(n)*Factorial(n+2)): n in [0..30] ]; // Vincenzo Librandi, Aug 19 2011
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CROSSREFS
| Cf. A000108.
Cf. A071716, A000778.
Sequence in context: A037028 A052919 * A167422 A060276 A025563 A110887
Adjacent sequences: A005804 A005805 A005806 * A005808 A005809 A005810
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| 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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