A006318 -id:A006318 - cp's OEIS frontend

A006319 Royal paths in a lattice (convolution of A006318).

1, 1, 4, 16, 68, 304, 1412, 6752, 33028, 164512, 831620, 4255728, 22004292, 114781008, 603308292, 3192216000, 16989553668, 90890869312, 488500827908, 2636405463248, 14281895003716, 77631035881072, 423282220216964, 2314491475510816, 12688544297945348

Offset: 0

N. J. A. Sloane

Number of peaks at level 1 in all Schröder paths of semilength n (n>=1). Example: a(2)=4 because in the six Schröder paths of semilength two, HH, H(UD), (UD)H, (UD)(UD), UHD and UUDD (where H=(2,0), U=(1,1), D=(1,-1)), we have four peaks at level 1 (shown between parentheses). - Emeric Deutsch, Dec 27 2003

a(n) = number of Schroder n-paths (subdiagonal paths of steps E = (1,0), N = (0,1), and D = (1,1) from the origin to (n,n) ) that start with an E step. For example, a(2) = 4 counts END, ENEN, EDN, EENN. - David Callan, May 15 2022

			a(4) = 68 since the top row of M^3 = (22, 22, 16, 8, 0, 0, 0, ...); where 68 = (22 + 22 + 16 + 8).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Axel Bacher, Directed and multi-directed animals on the square lattice with next nearest neighbor edges, arXiv preprint arXiv:1301.1365 [math.CO], 2013. See Q(t). - _N. J. A. Sloane_, Feb 14 2013
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 19.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Sai-nan Zheng and Sheng-liang Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.

First differences of A006318. Second diagonal of A033877.

[1] cat [&+[2/(k+2)*Binomial(n+k,k+1)*Binomial(n-1,k): k in [0..n]]: n in [1..25]]; // Vincenzo Librandi, Jul 22 2017

d[n_] := d[n] = Sum[(n - j)*Sum[d[i]d[j - i], {i, 0, j}], {j, 0, n-1}]; d[0] = 1; Table[d[n], {n, 0, 26}]
a[0] := 1; a[n_] := n Hypergeometric2F1[1 - n, n + 1, 3, -1];
Array[a, 25, 0] (* Peter Luschny, Jan 31 2020 *)

apply( {A006319(n)=!n+sum(k=0,n-1,binomial(n+k,k+1)*binomial(n-1,k)*2/(k+2))}, [0..30]) \\ M. F. Hasler, Jan 29 2020

def A006319_list(n) :
    D = [0]*(n+1); D[1] = 1
    b = True; h = 2; R = [1]
    for i in range(2*n-2) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1;
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-2]);
        b = not b
    return R
A006319_list(25) # Peter Luschny, Jun 03 2012

All listed terms satisfy the recurrence a(1) = 1 and, for n > 1, a(n) = 4*a(n-1) + Sum_{k=2..n-2} a(k)*a(n-k-1). - John W. Layman, Feb 23 2001
From Mario Catalani (mario.catalani(AT)unito.it), Jun 19 2003: (Start)
a(n) = Sum_{j=0..n} (n-j)*(Sum_{i=0..j} a(i)*a(j-i)) for n > 0, a(0)=1.
G.f.: A(x) = (1/(2x))((1-x)^2 - sqrt((1-x)^4 - 4*x*(1-x)^2)) (End)
a(n) = 0^n + Sum_{k=0..n-1} binomial(n+k, 2*k+1)*A000108(k+1). - Paul Barry, Feb 01 2009
G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z = x/(1-x)^2 (continued fraction); more generally g.f. C(x/(1-x)^2) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011
a(n) is the sum of top row terms in M^(n-1), M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  1, 1, 2, 0, 0, 0, ...
  1, 1, 1, 2, 0, 0, ...
  1, 1, 1, 1, 2, 0, ...
  1, 1, 1, 1, 1, 2, ...
  ... - Gary W. Adamson, Aug 23 2011
a(n) ~ 2^(1/4)*(3+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 09 2013
D-finite with recurrence: (n+1)*a(n) + (-7*n+4)*a(n-1) + (7*n-17)*a(n-2) + (-n+4)*a(n-3) = 0. - R. J. Mathar, Oct 16 2013
a(n) = Sum_{k=0..n} (2/(k+2))*binomial(n+k,k+1)*binomial(n-1,k) for n >= 1. - David Callan, Jul 21 2017
G.f. A(x) satisfies: A(x) = 1/(1 - Sum_{k>=1} k*x^k*A(x)). - Ilya Gutkovskiy, Apr 10 2018
From Peter Bala, Jan 28 2020: (Start)
a(n) = A006318(n) - A006318(n-1) for n >= 1.
(2*n-3)*(n+1)*a(n) = 12*(n-1)^2*a(n-1) - (2*n-1)*(n-3)*a(n-2) with a(1) = 1, a(2) = 4.
O.g.f. A(x) = (1 - x)*( (1 - x) - sqrt(1 - 6*x + x^2) )/(2*x) = (1 - x)*S(x) = 1 + x*S(x)^2, where S(x) is the o.g.f. for the large Schröder numbers A006318. (End)
a(n) = 0^n + n*hypergeom([1 - n, n + 1], [3], -1). - Peter Luschny, Jan 31 2020

A086616 Partial sums of the large Schroeder numbers (A006318).

Original entry on oeis.org

1, 3, 9, 31, 121, 515, 2321, 10879, 52465, 258563, 1296281, 6589727, 33887465, 175966211, 921353249, 4858956287, 25786112993, 137604139011, 737922992937, 3974647310111, 21493266631001, 116642921832963, 635074797251889, 3467998148181631, 18989465797056721, 104239408386028035

Offset: 0

Paul D. Hanna, Jul 24 2003

Row sums of triangle A086614. - Paul D. Hanna, Jul 24 2003

Hankel transform is A136577(n+1). - Paul Barry, Jun 03 2009

			a(1) = 2 + 1 = 3;
a(2) = 3 + 4 + 2 = 9;
a(3) = 4 + 10 + 12 + 5 = 31;
a(4) = 5 + 20 + 42 + 40 + 14 = 121.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009), #09.7.6.

Cf. A086614 (triangle), A086615 (antidiagonal sums).

Cf. A006318.

Table[SeriesCoefficient[(1-x-Sqrt[1-6*x+x^2])/(2*x*(1-x)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 14 2012 *)

x='x+O('x^66); Vec((1-x-sqrt(1-6*x+x^2))/(2*x*(1-x))) \\ Joerg Arndt, May 10 2013

# Generalized algorithm of L. Seidel
def A086616_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 2; R = []
    for i in range(2*n) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-1]); h += 1;
        b = not b
    return R
A086616_list(23) # Peter Luschny, Jun 02 2012

G.f.: A(x) = 1/(1 - x)^2 + x*A(x)^2.
a(1) = 1 and a(n) = n + Sum_{i=1..n-1} a(i)*a(n-i) for n >= 2. - Benoit Cloitre, Mar 16 2004
G.f.: (1 - x - sqrt(1 - 6*x + x^2))/(2*x*(1 - x)). Cf. A001003. - Ralf Stephan, Mar 23 2004
a(n) = Sum_{k=0..n} C(n+k+1, 2*k+1) * A000108(k). - Paul Barry, Jun 03 2009
Recurrence: (n+1)*a(n) = (7*n-2)*a(n-1) - (7*n-5)*a(n-2) + (n-2)*a(n-3). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ sqrt(24 + 17*sqrt(2))*(3 + 2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 14 2012
A(x) = 1/(1 - x)^2 * c(x/(1-x^2)), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Aug 29 2024

Name changed using a comment of Emeric Deutsch from Dec 20 2004. - Peter Luschny, Jun 03 2012

A333090 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of S(x)^n evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2x) is the o.g.f. of the Schröder numbers A006318.

Original entry on oeis.org

1, 3, 21, 183, 1729, 17003, 171237, 1752047, 18130433, 189218451, 1987916021, 20996253479, 222730436161, 2371369720827, 25325636818629, 271189884041183, 2910628489408513, 31302328583021091, 337241582882175189, 3639109029230457751, 39324814984207649729

Offset: 0

Peter Bala, Mar 22 2020

The sequence satisfies the Gauss congruences: a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) for all prime p and positive integers n and k.
We conjecture that the sequence satisfies the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all primes p >= 5 and positive integers n and k. Examples of these congruences are given below.
More generally, for each integer m, we conjecture that the sequence {a_m(n) : n >= 0}, defined by setting a_m(n) = the n-th order Taylor polynomial of S(x)^(m*n) evaluated at x = 1, satisfies the same supercongruences. See A333091 for m = 2 and A333092 for m = 3. For similarly defined sequences see A333093 through A333097.

			n-th order Taylor polynomial of S(x)^n:
  n = 0: S(x)^0 = 1 + O(x)
  n = 1: S(x)^1 = 1 + 2*x + O(x^2)
  n = 2: S(x)^2 = 1 + 4*x + 16*x^2 + O(x^3)
  n = 3: S(x)^3 = 1 + 6*x + 30*x^2 + 146*x^3 + O(x^4)
  n = 4: S(x)^4 = 1 + 8*x + 48*x^2 + 264*x^3 + 1408*x^4 + O(x^5)
Setting x = 1 gives a(0) = 1, a(1) = 1 + 2 = 3, a(2) = 1 + 4 + 16 = 21, a(3) = 1 + 6 + 30 + 146 = 183 and a(4) = 1 + 8 + 48 + 264 + 1408 = 1729.
The triangle of coefficients of the n-th order Taylor polynomial of S(x)^n, n >= 0, in descending powers of x begins
                                          row sums
  n = 0 |    1                                1
  n = 1 |    2    1                           3
  n = 2 |   16    4    1                     21
  n = 3 |  146   30    6   1                183
  n = 4 | 1408  264   48   8   1           1729
   ...
This is a Riordan array belonging to the Hitting time subgroup of the Riordan group. The first column sequence is [x^n]S(x)^n = A103885(n).
Examples of supercongruences:
a(13) - a(1) = 2371369720827 - 3 = (2^3)*(3^2)*(13^3)*83*180617 == 0 ( mod 13^3 ).
a(3*7) - a(3) = 425495386400395896971 - 183 = (2^2)*(7^3*)*19*47* 347287606554703 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 5894174066435445232142003 - 17003 = (2^3)*(3^4)*(5^6)*17* 41*101*5081*1627513421 == 0 ( mod 5^6 ).

Cf. A006318, A027307, A103885, A333091 through A333097, A372214, A372215.

S:= x -> (1/2)*(1-x-sqrt(1-6*x+x^2))/x:
G := (x,n) -> series(S(x)^n, x, 51):
seq(add(coeff(G(x, n), x, k), k = 0..n), n = 0..25);

Table[SeriesCoefficient[((1 + x)*(1 - Sqrt[1 - 4*x - 4*x^2])/(2*x))^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = [x^n] ( (1 + x)*S(x/(1 + x)) )^n.
O.g.f.: ( 1 + x*f'(x)/f(x) )/( 1 - x*f(x) ), where f(x) = 1 + 2*x + 10*x^2 + 66*x^3 + 498*x^4 + ... = (1/x)*Revert( x/S(x) ) is the o.g.f. of A027307.
Row sums of the Riordan array ( 1 + x*f'(x)/f(x), x*f(x) ) belonging to the Hitting time subgroup of the Riordan group.
a(n) ~ phi^(5*n+2) / (2*5^(3/4)*sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 28 2020

A186826 Riordan array (s(x),x*S(x)) where s(x) is the g.f. of the little Schroeder numbers A001003, and S(x) is the g.f. of the large Schroeder numbers A006318.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 11, 11, 5, 1, 45, 45, 23, 7, 1, 197, 197, 107, 39, 9, 1, 903, 903, 509, 205, 59, 11, 1, 4279, 4279, 2473, 1061, 347, 83, 13, 1, 20793, 20793, 12235, 5483, 1949, 541, 111, 15, 1, 103049, 103049, 61463, 28435, 10717, 3285, 795, 143, 17, 1, 518859, 518859, 312761, 148249, 58351, 19199, 5197, 1117, 179, 19, 1

Offset: 0

Paul Barry, Feb 27 2011

Reverse of A144944. Inverse of A186827.

			Triangle begins
       1;
       1,      1;
       3,      3,      1;
      11,     11,      5,      1;
      45,     45,     23,      7,     1;
     197,    197,    107,     39,     9,     1;
     903,    903,    509,    205,    59,    11,    1;
    4279,   4279,   2473,   1061,   347,    83,   13,    1;
   20793,  20793,  12235,   5483,  1949,   541,  111,   15,   1;
  103049, 103049,  61463,  28435, 10717,  3285,  795,  143,  17,  1;
  518859, 518859, 312761, 148249, 58351, 19199, 5197, 1117, 179, 19, 1;
Production matrix of this triangle begins
  1, 1;
  2, 2, 1;
  2, 2, 2, 1;
  2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 2, 1;
  2, 2, 2, 2, 2, 2, 2, 2, 2, 1;
For instance, 107=1*45+2*23+2*7+2*1.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A001003, A006318, A010683 (row sums), A144944 (row reverse), A186827 (inverse), A186828 (diagonal sums), A239204.

a186826 n k = a186826_tabl !! n !! k
a186826_row n = a186826_tabl !! n
a186826_tabl = map reverse a144944_tabl
-- Reinhard Zumkeller, May 11 2013

t[, 0]=1; t[p, p_]:= t[p, p]= t[p, p-1]; t[p_, q_]:= t[p, q]= t[p, q -1] + t[p-1, q] + t[p-1, q-1];
Table[t[p, q], {p,0,10}, {q,p,0,-1}]//Flatten (* Jean-François Alcover, Jul 16 2019 *)

@CachedFunction
def t(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (kA186826(n,k): return t(n+2,n-k)
flatten([[A186826(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 11 2023

Riordan array ((1+x+sqrt(1-6*x+x^2))/(4*x), (1-x-sqrt(1-6*x+x^2))/2).
Sum_{k=0..n} T(n,k) = A010683(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A186828(n).
R(n,k) = k*Sum_{i=0..n-k} (A001003(i)/(n-i))*Sum_{m=0..n-k-i} binomial(n-i,m)*binomial(2*(n-i)-m-k-1, n-i-1), k>0, R(n,0) = A001003(n). - Vladimir Kruchinin, Mar 09 2011
Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2). - G. C. Greubel, Mar 11 2023

A227505 Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

Original entry on oeis.org

1, 6, 31, 154, 763, 3808, 19197, 97772, 502749, 2607658, 13630635, 71743478, 379949431, 2023314980, 10828048409, 58206726936, 314157742457, 1701817879214, 9249717805207, 50427858276754, 275695956722547, 1511164724634440, 8302888160922965

Offset: 1

Johannes W. Meijer, Jul 15 2013

The terms of this sequence equal the Kn23 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).

Cf. A033877, A006603, A006318, A006319, A227504.

A227505 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if kA227505(n), n = 1..23);
A227505 := proc(n): A006603(n+3) - A006318(n+3) - A006319(n+2) end: A006603 := n ->  add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2))/(n-k+2), k= 1.. n/2+1): A006318 := n -> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): A006319 := proc(n): if n=0 then 1 else A006318(n) - A006318(n-1) fi: end: seq(A227505(n), n=1..23);

a(n) = sum(A033877(n-2*k+2,n-k+3), k=1..floor((n+1)/2)).

a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

A035011 A006318(n) - 1.

Original entry on oeis.org

0, 1, 5, 21, 89, 393, 1805, 8557, 41585, 206097, 1037717, 5293445, 27297737, 142078745, 745387037, 3937603037, 20927156705, 111818026017, 600318853925, 3236724317173

Offset: 0

N. J. A. Sloane

nonn

Number of occurrences of UD, UHD, UHHD, UHHHD, ... starting at level zero in all Schroeder paths of semilength n (i.e., lattice paths from (0,0) to (2n,0), with steps H=(2,0), U=(1,1) and D=(1,-1) and not going below the x-axis). Example: a(2) = 5 because in the six paths of semilength 2, namely HH, H(UD), (UD)H, (UHD), (UD)(UD), UUDD, we have 5 required occurrences (shown between parentheses). - Emeric Deutsch, Dec 28 2003

Fung Lam, Table of n, a(n) for n = 0..1300

Cf. A006318.

CoefficientList[Series[(1-4*x+x^2)/(2*x*(1-x))-Sqrt[1-6*x+x^2]/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 31 2014 *)

def A035011_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 2; R = []
    for i in range(2*n) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-2]); h += 1;
        b = not b
    return R
A035011_list(20) # Peter Luschny, Jun 03 2012

G.f.: (1-4*z+z^2)/(2*z*(1-z))-sqrt(1-6*z+z^2)/(2*z). - Emeric Deutsch, Dec 28 2003
Recurrence: (2*n^2 + 11*n + 12)*a(n+3) = (14*n^2 + 59*n + 60)*a(n+2) - (14*n^2 + 53*n + 48)*a(n+1) + (2*n^2 + 5*n)*a(n). - Ralf Stephan, Feb 11 2014
Asymptotics: a(n) ~ (3+2*sqrt(2))^n*(2^(1/4)+1/2^(1/4))/sqrt(2*Pi*n^3). - Fung Lam, Mar 31 2014
From Vaclav Kotesovec, Mar 31 2014: (Start)
Recurrence: (n+1)*a(n) = (8*n-3)*a(n-1) - 7*(2*n-3)*a(n-2) + (8*n-21)*a(n-3) - (n-4)*a(n-4).
Recurrence: (n+1)*(2*n-3)*a(n) = (2*n-1)*(7*n-9)*a(n-1) - (2*n-3)*(7*n-5)*a(n-2) + (n-3)*(2*n-1)*a(n-3).
(End)

A227504 Schroeder triangle sums: a(n) = A006603(n+1) - A006318(n+1).

Original entry on oeis.org

1, 4, 17, 74, 335, 1566, 7515, 36836, 183709, 929392, 4758477, 24611950, 128411643, 675051770, 3572165431, 19012868648, 101718917721, 546707554844, 2950563205705, 15983712882930, 86880753686279, 473710078493718, 2590187432233363, 14199709022579788

Offset: 1

Johannes W. Meijer, Jul 15 2013

The terms of this sequence equal the Kn22 sums, see A180662, of the Schroeder triangle A033877 (with offset 1 and n for columns and k for rows).

Cf. A033877, A006603, A006318, A227505.

A227504 := proc(n) local k, T; T := proc(n, k) option remember; if n=1 then return(1) fi; if kA227504(n), n = 1..24);
A227504 := proc(n): A006603(n+1) - A006318(n+1) end: A006603 := n -> add((k*add(binomial(n-k+2, i)*binomial(2*n-3*k-i+3, n-k+1), i= 0.. n-2*k+2)) / (n-k+2), k= 1.. n/2+1): A006318 := n -> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): seq(A227504(n), n=1..24);

a(n) = sum(A033877(n-2*k+2, n-k+2), k=1..floor((n+1)/2)).

a(n) = A006603(n+1) - A006318(n+1).

A333473 a(n) = [x^n] ( S(x/(1 + x)) )^n, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2x) is the o.g.f. of the large Schröder numbers A006318.

Original entry on oeis.org

1, 2, 12, 92, 752, 6352, 54768, 478928, 4231424, 37680320, 337622912, 3040354176, 27492359936, 249463806464, 2270319909632, 20714443816192, 189418898063360, 1735482632719360, 15928224355854336, 146414296847992832, 1347721096376573952, 12421053168197722112

Offset: 0

Peter Bala, Mar 23 2020

Let F(x) = 1 + f(1)*x + f(2)*x^2 + ... be a power series with integer coefficients. The associated sequence s(n) := [x^n] F(x)^n is known to satisfy the Gauss congruences: s(n*p^k) == s(n*p^(k-1)) ( mod p^(k) ) for any prime p and positive integers n and k. For certain power series F(x) we may get stronger congruences. Examples include F(x) = (1 + x)^2, F(x) = 1/(1 - x) and F(x) = c(x), where c(x) is the o.g.f. of the Catalan numbers A000108. The associated sequences (with some differences of offset) are A000984, A001700 and A025174, respectively.
Here we take F(x) = S(x/(1 + x)) = 1 + 2*x + 4*x^2 + 12*x^3 + 40*x^4 + 154*x^5 + 544*x^6 + ...(see A025227), where S(x) is the o.g.f. of the large Schröder numbers A006318. We conjecture that the associated sequence a(n) = [x^n] ( S(x/(1 + x)) )^n satisfies the congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) for prime p >= 5 and positive integers n and k. Cf. A333472.
More generally, we conjecture that for a positive integer r and integer s, the sequence a(r,s;n) := [x^(r*n)] ( S(x/(1 + x)) )^(s*n) also satisfies the above congruences.
Note the sequence b(n) := [x^n] ( S(x) )^n = A103885(n) appears to satisfy the stronger congruences b(n*p^k) == b(n*p^(k-1)) ( mod p^(3*k) ) for prime p >= 5 and positive integers n and k. The sequence d(n) := [x^n] ( (1 + x)*S(x/(1 + x)) )^n = A333090(n) also appears to satisfy the same congruences.

			Examples of congruences:
a(11) - a(1) = 3040354176 - 2 = 2*(11^2)*13*966419 == 0 ( mod 11^2 ).
a(3*7) - a(3) = 12421053168197722112 - 92 = (2^2)*(3^7)*5*(7^2)* 5795401942927 == 0 ( mod 7^2 ).
a(5^2) - a(5) = 90551762251592215396352 - 6352 = (2^4)*(5^4)*293* 30905038311123623 == 0 ( mod 5^4 ).

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Main diagonal of A378317.

Cf. A006318, A025227, A103885, A333090, A333472.

Sch := x -> (1/2)*(1-x-sqrt(1-6*x+x^2))/x:
G := x → Sch(x/(1+x));
H := (x, n) -> series(G(x)^n, x, 51):
seq(coeff(H(x, n), x, n), n = 0..25)

Table[SeriesCoefficient[((1 - Sqrt[1- 4*x - 4*x^2])/(2*x))^n, {x, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = sum(k=0, n, binomial(n, k)*binomial(n+2*k-1, 2*k)); \\ Seiichi Manyama, Nov 24 2024

a(n) = [x^n] ( (1 - sqrt(1- 4*x - 4*x^2))/(2*x) )^n.
a(n) ~ sqrt(((sqrt(2) + 1)^(2/3) + (sqrt(2) - 1)^(2/3) - 1)/3) * ((3*(71 + 8*sqrt(2))^(1/3) + 3*(71 - 8*sqrt(2))^(1/3) + 13))^n / (sqrt(Pi*n) * 2^(2*n+1)). - Vaclav Kotesovec, Mar 28 2020
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+2*k-1,2*k). - Seiichi Manyama, Nov 24 2024

A108891 Triangle read by rows: T(n,k) = number of Schroeder (or royal) n-paths (A006318) containing k returns to the diagonal y=x. (A northeast step lying on y=x contributes a return.)

Original entry on oeis.org

2, 2, 4, 6, 8, 8, 22, 28, 24, 16, 90, 112, 96, 64, 32, 394, 484, 416, 288, 160, 64, 1806, 2200, 1896, 1344, 800, 384, 128, 8558, 10364, 8952, 6448, 4000, 2112, 896, 256, 41586, 50144, 43392, 31616, 20160, 11264, 5376, 2048, 512

Offset: 1

David Callan, Jul 25 2005

			Table begins
  n\k  1    2    3    4    5    6
  -------------------------------
  1 |  2
  2 |  2    4
  3 |  6    8    8
  4 | 22   28   24   16
  5 | 90  112   96   64   32
  6 |394  484  416  288  160   64
The paths DD, END, DEN, ENEN each have 2 returns (E=east, N=north, D=northeast); so T(2,2)=4.
From _Philippe Deléham_, Nov 02 2013: (Start)
Triangle (0, 1, 2, 1, 2, 1, 2, ...) DELTA (1, 0, 0, 0, ...) begins:
  1;
  0,   2;
  0,   2,   4;
  0,   6,   8,   8;
  0,  22,  28,  24,  16;
  0,  90, 112,  96,  64,  32;
  0, 394, 484, 416, 288, 160,  64; (End)

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
Scott Balchin, Ethan MacBrough, and Kyle Ormsby, The combinatorics of N_oo operads for C_qp^n and D_p^n, arXiv:2209.06992 [math.AT], 2022.

Row sums are the large Schroeder numbers A006318. Column k=1 is twice the little Schroeder numbers A001003. The main diagonal consists of powers of 2, A000079. The first subdiagonal is A036289. The analogous Catalan triangle is A009766 (with rows reversed).

T[n_, k_] := (-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, 2]; Table[T[n - 1, k - 1]*2^k, {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Sep 21 2022, after Peter Luschny at A104219 *)

Column k is the k-fold convolution of column 1.
T(n, k) = A104219(n-1, k-1)*2^k. - Philippe Deléham, Jul 31 2005
Triangle T(n,k), 1 <= k <= n, read by rows given by (0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 02 2013

A138462 A bisection of A006318.

Original entry on oeis.org

1, 6, 90, 1806, 41586, 1037718, 27297738, 745387038, 20927156706, 600318853926, 17518619320890, 518431875418926, 15521467648875090, 469286147871837366, 14308406109097843626, 439442782615614361662, 13582285614213903189954, 422158527806921249683014

Offset: 0

N. J. A. Sloane, May 08 2008

Colin Barker, Table of n, a(n) for n = 0..300

a(n):=sum(binomial(2*n+1,i+1)*binomial(2*n+i,i),i,0,2*n)/(2*n+1); /* Vladimir Kruchinin, Apr 02 2017 */

a(n) = sum(i=0, 2*n, binomial(2*n+1,i+1)*binomial(2*n+i,i))/(2*n+1) \\ Colin Barker, Dec 28 2017

a(n) = A006318(2n).
a(n) = Sum_{i=0..2n} (C(2*n+1,i+1)*C(2*n+i,i))/(2*n+1). - Vladimir Kruchinin, Apr 02 2017
D-finite with recurrence +n*(2*n+1)*a(n) +(-70*n^2+73*n-15)*a(n-1) +(70*n^2-277*n+270)*a(n-2) -(2*n-5)*(n-3)*a(n-3)=0. - R. J. Mathar, Mar 25 2024

cp's OEIS Frontend

A006319 Royal paths in a lattice (convolution of A006318).

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A086616 Partial sums of the large Schroeder numbers (A006318).

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A333090 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of S(x)^n evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the Schröder numbers A006318.

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A186826 Riordan array (s(x),x*S(x)) where s(x) is the g.f. of the little Schroeder numbers A001003, and S(x) is the g.f. of the large Schroeder numbers A006318.

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A227505 Schroeder triangle sums: a(n) = A006603(n+3) - A006318(n+3) - A006319(n+2).

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A035011 A006318(n) - 1.

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A227504 Schroeder triangle sums: a(n) = A006603(n+1) - A006318(n+1).

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A333090 a(n) is equal to the n-th order Taylor polynomial (centered at 0) of S(x)^n evaluated at x = 1, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2x) is the o.g.f. of the Schröder numbers A006318.

A333473 a(n) = [x^n] ( S(x/(1 + x)) )^n, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2x) is the o.g.f. of the large Schröder numbers A006318.