A297705 -id:A297705 - cp's OEIS frontend

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

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

A297899 Triangle read by rows, T(n, k) = binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 5, 1, 45, 10, 1, 505, 115, 15, 1, 6345, 1460, 210, 20, 1, 85405, 19765, 2990, 330, 25, 1, 1204245, 279710, 43635, 5220, 475, 30, 1, 17558705, 4088615, 651165, 81955, 8275, 645, 35, 1, 262577745, 61254760, 9901860, 1290520, 139350, 12280, 840, 40, 1

Offset: 0

Peter Luschny, Jan 08 2018

			Triangle starts:
[0]       1
[1]       5,      1
[2]      45,     10,     1
[3]     505,    115,    15,    1
[4]    6345,   1460,   210,   20,   1
[5]   85405,  19765,  2990,  330,  25,  1
[6] 1204245, 279710, 43635, 5220, 475, 30, 1

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

T(n, 0) = A133305(n). Row sums are A297705, alternating row sums are A131765.

Cf. A103209.

T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
T[n_, k_] := Sum[4^(j - k)*(k + 1)*Binomial[n + j - k, 2*j - k]*Binomial[2*j - k, j - k]/(j + 1), {j, k, n}];
Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* Detlef Meya, Jan 15 2024 *)

T(n,k) = sum(j = k, n, 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1)) \\ Andrew Howroyd, Jan 15 2024

T(n, k) = Sum_{j = k..n} 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1). - Detlef Meya, Jan 15 2024

cp's OEIS Frontend

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

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