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 A030238 Backwards shallow diagonal sums of Catalan triangle A009766. 8
 1, 1, 3, 7, 20, 59, 184, 593, 1964, 6642, 22845, 79667, 281037, 1001092, 3595865, 13009673, 47366251, 173415176, 638044203, 2357941142, 8748646386, 32576869203, 121701491701, 456012458965, 1713339737086 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of linear forests of planted planar trees with n nodes (Christian G. Bower). 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 Hankel transform is A056520. - Paul Barry, Oct 16 2007 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA INVERT transform of 1, 2, 2, 5, 14, 42, 132, ... (cf. A000108). 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 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 G.f.: (1-sqrt(1-4*z))/(z*(2-z+z*sqrt(1-4*z)). - Emeric Deutsch, Mar 01 2007 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 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 a(n) = Sum_{k=0..n} A000108(k)*A132364(n-k). - Philippe Deléham, Feb 27 2013 a(n) ~ 2^(2*n+6) / (49 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014 MAPLE 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 MATHEMATICA Sum[ triangle[ n-k, (n-k)-(k-1) ], {k, 1, Floor[ (n+1)/2 ]} ] CoefficientList[Series[(1-Sqrt[1-4*x])/x/(2-x+x*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *) CROSSREFS Cf. A000108, A009766, A127158, A132364. Sequence in context: A238124 A129429 A084204 * A132364 A110490 A132868 Adjacent sequences: A030235 A030236 A030237 * A030239 A030240 A030241 KEYWORD nonn AUTHOR EXTENSIONS More terms from Christian G. Bower, Apr 15 1998 STATUS approved

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Last modified December 9 17:12 EST 2022. Contains 358702 sequences. (Running on oeis4.)