A030238 Backwards shallow diagonal sums of Catalan triangle A009766.

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

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

S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)

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

Wouter Meeussen

Extensions

More terms from Christian G. Bower, Apr 15 1998

Status

approved