Search: id:A000958
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%I A000958 M2748 N1104
%S A000958 1,1,3,8,24,75,243,808,2742,9458,33062,116868,417022,1500159,
%T A000958 5434563,19808976,72596742,267343374,988779258,3671302176,13679542632,
%U A000958 51134644014,191703766638,720629997168,2715610275804,10256844598900
%N A000958 Number of ordered rooted trees with n edges having root of odd degree.
%C A000958 a(n) = number of Dyck n-paths containing no peak at height 2 before the 
               first return to ground level. Example: a(3)=3 counts UUUDDD, UDUUDD, 
               UDUDUD. - David Callan (callan(AT)stat.wisc.edu), Jun 07 2006
%C A000958 Also number of order trees with n edges and having no even-length branches 
               starting at the root. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Mar 02 2007
%C A000958 Convolution of the Catalan sequence 1,1,2,5,14,42,... (A000108) and the 
               Fine sequence 1,0,1,2,6,18,... (A000957). a(n)=A127541(n,0). - Emeric 
               Deutsch (deutsch(AT)duke.poly.edu), Mar 02 2007
%C A000958 The Catalan transform of A008619. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Nov 06 2008]
%C A000958 Hankel transform is F(2n+1). [From Paul Barry (pbarry(AT)wit.ie), Dec 
               01 2008]
%C A000958 Starting with offset 2 = iterates of M * [1,1,0,0,0,...] where M = a 
               tridiagonal matrix with [0,2,2,2,...] in the main diagonal and [1,
               1,1,...] in the super and subdiagonals. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jan 09 2009]
%C A000958 Equals INVERT transform of A032357 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Apr 10 2009]
%D A000958 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000958 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000958 E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 
               241 (2001), 241-265.
%D A000958 Fine, Terrence; Extrapolation when very little is known about the source. 
               Information and Control 16 (1970), 331-359.
%H A000958 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%F A000958 Equals A000957(n) + A000957(n+1).
%F A000958 G.f.:=(1-x-(1+x)*sqrt(1-4*x))/(2x(x+2)); - Paul Barry (pbarry(AT)wit.ie), 
               Jan 26 2007
%F A000958 G.f.=zC/(1-z^2*C^2), where C=(1-sqrt(1-4z))/(2z) is the Catalan function. 
               - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 02 2007
%F A000958 a(n+1)=Sum_{k, 0<=k<=[n/2]}A039599(n-k,k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 13 2007
%F A000958 a(n) = (-1/2)^n*(-2-5*sum((-8)^k*GAMMA(1/2+k)*(4/5+k)/(sqrt(Pi)*GAMMA(k+3)),
               k = 1 .. -1+n)) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 
               11 2009]
%F A000958 a(n)+a(n+1)=A135339(n+1). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 02 2009]
%p A000958 g:=(1-x-(1+x)*sqrt(1-4*x))/2/x/(x+2): gser:=series(g,x=0,30): seq(coeff(gser,
               x,n),n=1..26); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 02 
               2007
%Y A000958 A column of A065602.
%Y A000958 Cf. A127541, A127539, A000108, A000957.
%Y A000958 A032357 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 10 2009]
%Y A000958 Sequence in context: A006365 A046919 A046342 this_sequence A148782 A148783 
               A084205
%Y A000958 Adjacent sequences: A000955 A000956 A000957 this_sequence A000959 A000960 
               A000961
%K A000958 nonn,easy,new
%O A000958 1,3
%A A000958 N. J. A. Sloane (njas(AT)research.att.com).

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