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%I #47 Apr 12 2023 11:07:55
%S 1,1,3,7,20,59,184,593,1964,6642,22845,79667,281037,1001092,3595865,
%T 13009673,47366251,173415176,638044203,2357941142,8748646386,
%U 32576869203,121701491701,456012458965,1713339737086
%N Backwards shallow diagonal sums of Catalan triangle A009766.
%C Number of linear forests of planted planar trees with n nodes (Christian G. Bower).
%C Number of ordered trees with n+2 edges and having no branches of length 1 starting from the root. Example: a(1)=1 because the only ordered tree with 3 edges having no branch of length 1 starting from the root is the path tree of length 3. a(n) = A127158(n+2,0). - _Emeric Deutsch_, Mar 01 2007
%C Hankel transform is A056520. - _Paul Barry_, Oct 16 2007
%H Vincenzo Librandi, <a href="/A030238/b030238.txt">Table of n, a(n) for n = 0..200</a>
%H S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017)
%F INVERT transform of 1, 2, 2, 5, 14, 42, 132, ... (cf. A000108).
%F a(n) = Sum_{k=0..floor(n/2)} (k+1)*binomial(2*n-3*k+1, n-k+1)/(2*n-3*k+1)}. Diagonal sums of A033184. - _Paul Barry_, Jun 22 2004
%F a(n) = Sum_{k=0..floor(n/2)} (k+1)*binomial(2*n-3*k, n-k)/(n-k+1). - _Paul Barry_, Feb 02 2005
%F G.f.: (1-sqrt(1-4*z))/(z*(2-z+z*sqrt(1-4*z)). - _Emeric Deutsch_, Mar 01 2007
%F G.f.: c(z)/(1-z^2*c(z)) where c(z) = (1-sqrt(1-4*z))/(2*z). - _Ira M. Gessel_, Sep 21 2020
%F D-finite with recurrence: (n+1)*a(n) + (-5*n+1)*a(n-1) + 2*(2*n-1)*a(n-2) + (n+1)*a(n-3) + 2*(-2*n+1)*a(n-4) = 0. - _R. J. Mathar_, Nov 30 2012
%F a(n) = Sum_{k=0..n} A000108(k)*A132364(n-k). - _Philippe Deléham_, Feb 27 2013
%F a(n) ~ 2^(2*n+6) / (49 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2014
%p g:=(1-sqrt(1-4*z))/z/(2-z+z*sqrt(1-4*z)): gser:=series(g,z=0,30): seq(coeff(gser,z,n),n=0..25); # _Emeric Deutsch_, Mar 01 2007
%t Sum[ triangle[ n-k, (n-k)-(k-1) ], {k, 1, Floor[ (n+1)/2 ]} ]
%t CoefficientList[Series[(1-Sqrt[1-4*x])/x/(2-x+x*Sqrt[1-4*x]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%Y Cf. A000108, A009766, A127158, A132364.
%K nonn
%O 0,3
%A _Wouter Meeussen_
%E More terms from _Christian G. Bower_, Apr 15 1998
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