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, 6453584646837, 24364925260024, 92185136438942

Offset: 0

Author

Wouter Meeussen

Keywords

• nonn

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)
• Taras Goy and Mark Shattuck, Hessenberg-Toeplitz Matrix Determinants with Schröder and Fine Number Entries, Carpathian Math. Publ., Vol. 15 (2023), No. 2, 420-436. See Theorem 3.

Crossrefs

Cf. A000108, A009766, A127158, A132364.

Programs

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 *)

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

Extensions

More terms from Christian G. Bower, Apr 15 1998