Search: id:A014307
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%I A014307
%S A014307 1,1,2,7,35,226,1787,16717,180560,2211181,30273047,458186752,
%T A014307 7596317885,136907048461,2665084902482,55726440112987,1245661569161135,
%U A014307 29642264728189066,748158516941653967,19962900431638852297
%N A014307 Expansion of sqrt( exp(x) / ( 2 - exp(x) )).
%C A014307 Tha Hankel transform of this sequence is A121835 . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Aug 31 2006
%C A014307 a(n) is the moment of order (n-1) for the discrete measure associated 
               to the weight rho(j+1/2)=2^(j+1/2)/(Pi*binomial(2j+1,j+1/2)), with 
               j integral. So we have a(n)=sum((j+1/2)^(n-1)*rho(j+1/2),j=0..infinity). 
               [From roland groux (roland.groux(AT)orange.fr), Jan 05 2009]
%D A014307 M. Klazar, Twelve countings with rooted plane trees, European Journal 
               of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
%F A014307 Recurrence : a(n+1) = 1 + sum { j=1, n, (-1+binomial(n+1, j))*a(n) } 
               - Jon Perry (perry(AT)globalnet.co.uk), Apr 25 2005
%F A014307 The Hankel transform of this sequence is A121835 . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Aug 31 2006
%F A014307 E.g.f. A(x) satisfies: A(x) = 1 + integral( A(x)^3 * exp(-x) ). - Paul 
               D. Hanna (pauldhanna(AT)juno.com), Jan 24 2008
%o A014307 (PARI) {a(n)=n!*polcoeff((exp(x +x*O(x^n))/(2-exp(x +x*O(x^n))))^(1/2),
               n)} (PARI) /* As solution to integral equation: */ {a(n)=local(A=1+x+x*O(x^n));
               for(i=0,n,A=1+intformal(A^3*exp(-x+x*O(x^n))));n!*polcoeff(A,n)} 
               - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 24 2008
%Y A014307 Cf. A000110.
%Y A014307 Cf. variants: A136727, A136728, A136729.
%Y A014307 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 20 
               2009: (Start)
%Y A014307 Equals row sums of triangle A156920 (row sums (n) = a(n+1))
%Y A014307 (End)
%Y A014307 Sequence in context: A043546 A080831 A006947 this_sequence A000154 A003713 
               A058129
%Y A014307 Adjacent sequences: A014304 A014305 A014306 this_sequence A014308 A014309 
               A014310
%K A014307 nonn
%O A014307 0,3
%A A014307 N. J. A. Sloane (njas(AT)research.att.com).

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