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Search: id:A005807
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| A005807 |
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Sum of adjacent Catalan numbers.
(Formerly M0850)
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+0
8
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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; listen)
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OFFSET
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0,1
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REFERENCES
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D. E. Knuth, personal communication.
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LINKS
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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
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 431
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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(1-4x))/(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
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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PROGRAM
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(PARI) a(n)=if(n<0, 0, binomial(2*n, n)*(5*n+4)/(n+1)/(n+2))
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CROSSREFS
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Cf. A000108.
Cf. A071716, A000778.
Sequence in context: A033844 A037028 A052919 this_sequence A060276 A025563 A110887
Adjacent sequences: A005804 A005805 A005806 this_sequence A005808 A005809 A005810
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KEYWORD
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nonn
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AUTHOR
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njas
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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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