0,1

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. E. Knuth, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

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).

a[n_]:=Binomial[2*n, n]*(5*n+4)/(n+1)/(n+2); [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 13 2008]

(PARI) a(n)=if(n<0, 0, binomial(2*n, n)*(5*n+4)/(n+1)/(n+2))

(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]

Cf. A000108.

Cf. A071716, A000778.

Sequence in context: A033844 A037028 A052919 this_sequence A167422 A060276 A025563

Adjacent sequences: A005804 A005805 A005806 this_sequence A005808 A005809 A005810

nonn

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Joe Keane (jgk(AT)jgk.org), Feb 08 2000

Asymptotic series corrected and extended by Michael Somos, Sep 11 2003.