login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Backwards shallow diagonal sums of Catalan triangle A009766.
8

%I #52 Aug 06 2024 21:16:52

%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