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A030238 Backwards shallow diagonal sums of Catalan triangle A009766. 7
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

S. B. Ekhad, 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(2n-3k+1, n-k+1)/(2n-3k+1)}. Diagonal sums of A033184. - Paul Barry, Jun 22 2004

a(n)=sum{k=0..floor(n/2), (k+1)binomial(2n-3k, n-k)/(n-k+1)} - Paul Barry, Feb 02 2005

G.f.=[1-sqrt(1-4z)]/[z(2-z+z*sqrt(1-4z)]. - Emeric Deutsch, Mar 01 2007

Conjecture: (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<=k<=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. A127158, A132364.

Sequence in context: A129429 A084204 * A132364 A110490 A132868 A056783

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

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Last modified February 18 13:07 EST 2018. Contains 299322 sequences. (Running on oeis4.)