A114709 -id:A114709 - cp's OEIS frontend

A186940 Diagonal sums of number triangle A114709.

1, 0, 3, 6, 31, 126, 589, 2772, 13485, 66780, 336207, 1714698, 8841627, 46015002, 241394073, 1275137448, 6776728825, 36208438488, 194388488155, 1048061471886, 5672504958327, 30808982057046, 167864115588325, 917271225518076, 5025659929354981

Offset: 0

Paul Barry, Mar 01 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A114709.

Hankel transform is A186941.

Rest[CoefficientList[Series[(1+3*x-2*x^2-Sqrt[1-6*x+x^2])/(2*(3+x-3*x^2+x^3)), {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 01 2014 *)

G.f. (for offset 1): (1+3*x-2*x^2-sqrt(1-6*x+x^2))/(2*(3+x-3*x^2+x^3)).
Conjecture: 3*(n+1)*a(n) +(10-17*n)*a(n-1) -6*(n+1)*a(n-2) +10*(2*n-1)*a(n-3) +9*(1-n)*a(n-4) +(n-2)*a(n-5) =0. - R. J. Mathar, Nov 17 2011
a(n) ~ sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^(n+2) / (36 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014

More terms from Vincenzo Librandi, Feb 14 2014

A114710 Number of hill-free Schroeder paths of length 2n that have no horizontal steps on the x-axis.

Original entry on oeis.org

1, 0, 2, 6, 26, 114, 526, 2502, 12194, 60570, 305526, 1560798, 8058714, 41987106, 220470942, 1165553718, 6198683090, 33140219946, 178012804678, 960232902606, 5199384505226, 28250295397170, 153977094874862, 841656387060006

Offset: 0

Emeric Deutsch, Dec 26 2005

A Schroeder path of length 2n is a lattice path from (0, 0) to (2n, 0) consisting of U = (1,1), D = (1,-1) and H = (2,0) steps and never going below the x-axis. A hill is a peak at height 1.

Hankel transform is 2^C(n+1,2) (A006125(n+1)). Hankel transform of a(n+1) is (2-2^(n+1))*2^C(n+1,2). - Paul Barry, Oct 31 2008

			a(3) = 6 because we have UHHD, UHUDD, UUDHD, UUDUDD, UUHDD and UUUDDD.

Michael De Vlieger, Table of n, a(n) for n = 0..1000
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.

Column 0 of A114709.

Cf. A001003, A104219, A297705.

G:=2/(1+3*z+sqrt(1-6*z+z^2)): Gser:=series(G,z=0,32):
1,seq(coeff(Gser,z^n),n=1..27);
# Alternative:
a := proc(n) option remember; if n < 3 then return [1, 0, 2, 6][n+1] fi;
((4 - 2*n)*a(n-3) + (16*n - 11)*a(n-1) + 9*n*a(n-2))/(3*n + 3) end:
seq(a(n), n = 0..23); # Peter Luschny, Nov 10 2022

A114710[n_] := (-1)^n Sum[Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 2], {k, 0, n}]; Table[A114710[n], {n, 0, 23}] (* Peter Luschny, Jan 08 2018 *)
InverseInvertTransform[ser_, n_] := CoefficientList[Series[ser/(1 + x ser), {x, 0, n}], x]; LittleSchroeder := (1 + x - Sqrt[1 - 6 x + x^2])/(4 x); (* A001003 *)
InverseInvertTransform[LittleSchroeder, 23] (* Peter Luschny, Jan 10 2019 *)

G.f.: A(x) = 2/(1+3*x+sqrt(1-6*x+x^2)).
D-finite with recurrence 3*(n+1)*a(n) +(11-16*n)*a(n-1) -9*n*a(n-2) +2*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 07 2012
G.f.: 1/(Q(0) + 2*x) where Q(k) = 1 + k*(1-x) - x - x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(n) = (-1)^n*Sum_{k=0..n} binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], 2). - Peter Luschny, Jan 08 2018
O.g.f. A(x) = 1/x * series reversion of x*(1 - 3*x)/((1 - x)*(1 - 2*x)). Cf. A297705. - Peter Bala, Nov 08 2022
a(n) ~ (9 + 4*sqrt(2)) * (1 + sqrt(2))^(2*n + 1) / (49 * sqrt(Pi) * 2^(3/4) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2022

cp's OEIS Frontend

A186940 Diagonal sums of number triangle A114709.

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