Search: id:A005807
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%I A005807 M0850
%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)). 
               [From Paul Barry (pbarry(AT)wit.ie), 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 Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, <a href="http:/
               /www.cs.uwaterloo.ca/journals/JIS/index.html">Catalan Numbers, the 
               Hankel Transform and Fibonacci Numbers</a>, Journal of Integer Sequences, 
               Vol. 5 (2002), Article 02.1.3
%H A005807 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=431">
               Encyclopedia of Combinatorial Structures 431</a>
%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(1-4x))/(2x^2)=(4+2x)/(1-x+(1+x)sqrt(1-4x)). a(n)(n+2)(5n-1)=a(n-1)2(2n-1)(5n+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).
%t A005807 a[n_]:=Binomial[2*n, n]*(5*n+4)/(n+1)/(n+2); [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Dec 13 2008]
%o A005807 (PARI) a(n)=if(n<0,0,binomial(2*n,n)*(5*n+4)/(n+1)/(n+2))
%o A005807 (Other) sage: [catalan_number(i)+catalan_number(i+1) for i in xrange(0,
               25)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 17 
               2009]
%Y A005807 Cf. A000108.
%Y A005807 Cf. A071716, A000778.
%Y A005807 Sequence in context: A033844 A037028 A052919 this_sequence A167422 A060276 
               A025563
%Y A005807 Adjacent sequences: A005804 A005805 A005806 this_sequence A005808 A005809 
               A005810
%K A005807 nonn
%O A005807 0,1
%A A005807 N. J. A. Sloane (njas(AT)research.att.com).
%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.

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