A253487 -id:A253487 - cp's OEIS frontend

A110609 a(n) = n * binomial(2*n, n-1).

0, 1, 8, 45, 224, 1050, 4752, 21021, 91520, 393822, 1679600, 7113106, 29953728, 125550100, 524190240, 2181340125, 9051563520, 37467344310, 154754938800, 637982011590, 2625648168000, 10789623755820, 44277560801760, 181478535620850, 742984788858624, 3038716500907500

Offset: 0

Paul Barry, Jul 30 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.

Cf. A000108, A002390, A002736, A253487.

Column k=1 of A110608.

[0] cat [((4*n+4)*(2*n+1)*Binomial(2*n, n)/(n+2))/2: n in [0..25]]; // Vincenzo Librandi, Jan 09 2015

with(combinat):with(combstruct):a[0]:=0:for n from 1 to 30 do a[n]:=sum((count(Composition(n*2+1),size=n)),j=1..n) od: seq(a[n], n=0..22); # Zerinvary Lajos, May 09 2007
a:=n->sum(sum(binomial(2*n,n)/(n+1), j=1..n),k=1..n): seq(a(n), n=0..22); # Zerinvary Lajos, May 09 2007
series(simplify(x*diff(x*diff((1-sqrt(1-4*x))/(2*x), x), x)), x, 20):
seq(coeff(%, x, k), k=0..18); # Karol A. Penson, Apr 25 2025

Table[CatalanNumber[n]*n^2, {n, 0, 22}] (* Zerinvary Lajos, Jul 08 2009 *)
CoefficientList[Series[x (1 / x^2 - (1 - 6 x + 4 x^2) / ((1 - 4 x)^(3/2) x^2)) / 2, {x, 0, 30}], x] (* Vincenzo Librandi, Jan 09 2015 *)

for(n=0,25, print1(n*binomial(2*n,n-1), ", ")) \\ G. C. Greubel, Sep 01 2017

a(n) = n^2*binomial(2*n, n)/(n+1) = n^2*A000108(n) = A002736(n)/(n+1).
G.f.: -(2*x*(2*x+2*sqrt(1-4*x)-3) - sqrt(1-4*x) + 1)/(2*sqrt((1 - 4*x)^3)*x). - Marco A. Cisneros Guevara, Jul 23 2011; amended by Georg Fischer, Apr 09 2020
(n+1)*(10*n-7)*a(n)+2*n*(5*n-88)*a(n-1) -4*(25*n-22)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 07 2012
From Ilya Gutkovskiy, Jan 20 2017: (Start)
E.g.f.: x*(BesselI(0,2*x) + 2*BesselI(1,2*x) + BesselI(2,2*x))*exp(2*x).
a(n) ~ 4^n*sqrt(n)/sqrt(Pi).
Sum_{n>=1} 1/a(n) = Pi*(2*sqrt(3) + Pi)/18 = 1.152911143694148... (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2/sqrt(5))*log(phi) + 2*log(phi)^2, where log(phi) = A002390. - Amiram Eldar, Feb 20 2021
G.f.: (x*(d/dx))^2 [g.f. of A000108]. - Karol A. Penson, Apr 25 2025

A351585 Triangle read by rows: T(n,k) = A(k,n-k), 1 <= k < n, 2 <= n, where A(m,n) is the number of distinct strings consisting of one X, m+n-1 Y's and m+n-1 Z's in which the X lies to the right of at least m Y's and at least m Z's.

Original entry on oeis.org

2, 16, 6, 90, 52, 20, 448, 306, 180, 70, 2100, 1568, 1086, 644, 252, 9504, 7500, 5664, 3948, 2352, 924, 42042, 34452, 27450, 20840, 14580, 8712, 3432, 183040, 154154, 127380, 101950, 77640, 54450, 32604, 12870, 787644, 677248, 574574, 476652, 382510, 291896, 205062, 122980, 48620

Offset: 2

Christopher J. Fewster, Feb 14 2022

The general string enumeration problem of counting strings with k+k'-1 X's, m+m' Y's and n+n' Z's in which the k'th X is placed after at least m of the Y's and n of the Z's may be expressed in terms of an integral of incomplete Beta functions and evaluated in terms of Kampe de Feriet functions (see Connor & Fewster, 2022). Other special cases include A351583 and A351584.

			Triangle starts:
     2;
    16,    6;
    90,   52,   20;
   448,  306,  180,  70;
  2100, 1568, 1086, 644, 252;
  ...

Stephen B. Connor and Christopher J. Fewster, Integrals of incomplete beta functions, with applications to order statistics, random walks and string enumeration, Brazilian Journal of Probability and Statistics 2022, Vol. 36, No. 1, 185-198; arXiv version, arXiv:2104.12216 [math.CA], 2021.

Cf. A000984, A253487, A351583, A351584.

T:=(n,k)->(n - k)*binomial(2*n - 1, n) - 2*k*(n - k)*binomial(2*k - 1, k)*binomial(2*n - 2*k - 1, n - k)/n; [seq(seq(T(n,k),k=1..n-1),n=2..10)];

t[n_,k_]:=(n-k)*Binomial[2*n-1,n]-(2*k*(n-k)/n)*Binomial[2*k-1,k]*Binomial[2*(n-k)-1,n-k]; Table[t[n,k],{n,2,10},{k,1,n-1}]

T(n+2,1) = A(1,n) = 2*n*binomial(2*n,n-1) = A253487(n-1).
T(m+1,m) = A(m,1) = binomial(2*m,m) = A000984(m) [central binomial coefficients].
T(n,k) = (n - k)*binomial(2*n - 1, n) - 2*k*(n - k)*binomial(2*k - 1, k)*binomial(2*n - 2*k - 1, n - k)/n. See Connor & Fewster (2022).

A380119 Triangle read by rows: T(n, k) is the number of walks of length 2*n on the N X N grid with unit steps in all four directions (NSWE) starting at (0, 0). k is the common value of the x- and the y-coordinate of the endpoint of the walk.

Original entry on oeis.org

1, 2, 2, 10, 16, 6, 70, 140, 90, 20, 588, 1344, 1134, 448, 70, 5544, 13860, 13860, 7392, 2100, 252, 56628, 151008, 169884, 109824, 42900, 9504, 924, 613470, 1717716, 2108106, 1561560, 750750, 231660, 42042, 3432, 6952660, 20225920, 26546520, 21781760, 12155000, 4667520, 1191190, 183040, 12870

Offset: 0

Peter Luschny, Jan 19 2025

			The triangle starts:
  [0] [      1]
  [1] [      2,        2]
  [2] [     10,       16,        6]
  [3] [     70,      140,       90,      20]
  [4] [    588,     1344,     1134,      448,       70]
  [5] [   5544,    13860,    13860,     7392,     2100,     252]
  [6] [  56628,   151008,   169884,   109824,    42900,    9504,     924]
  [7] [ 613470,  1717716,  2108106,  1561560,   750750,  231660,   42042,   3432]
  [8] [6952660, 20225920, 26546520, 21781760, 12155000, 4667520, 1191190, 183040, 12870]
.
For n = 2 the walks depending on the x-coordinate of the endpoint are:
W(x=0) = {NNSS,NSNS,NSWE,NWSE,NWES,WNSE,WNES,WWEE,WENS,WEWE},
W(x=1) = {NNSW,NNWS,NSNW,NSWN,NWNS,NWSN,NWWE,NWEW,WNNS,WNSN,WNWE,WNEW,WWNE,WWEN,WENW,WEWN},
W(x=2) = {NNWW,NWNW,NWWN,WNNW,WNWN,WWNN}.

M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.
Richard K. Guy, Christian Krattenthaler and Bruce E. Sagan, Lattice paths, reflections, & dimension-changing bijections, Ars Combin. 34 (1992), 3-15.

Related triangles: A380120.

Cf. A005568 (column 0), A000984 (main diagonal), A253487 (sub diagonal), A151403 (row sums).

from dataclasses import dataclass
@dataclass
class Walk: s: str = ""; x: int = 0; y: int = 0
def Trow(n: int) -> list[int]:
    W = [Walk()]
    row = [0] * (n + 1)
    for w in W:
        if len(w.s) == 2*n:
            if w.x == w.y: row[w.y] += 1
        else:
            for s in "NSWE":
                x = y = 0
                match s:
                    case "W": x =  1
                    case "E": x = -1
                    case "N": y =  1
                    case "S": y = -1
                    case _  : pass
                if (w.y + y >= 0) and (w.x + x >= 0):
                    W.append(Walk(w.s + s, w.x + x, w.y + y))
    return row
for n in range(6): print(Trow(n))

cp's OEIS Frontend

A110609 a(n) = n * binomial(2*n, n-1).

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A351585 Triangle read by rows: T(n,k) = A(k,n-k), 1 <= k < n, 2 <= n, where A(m,n) is the number of distinct strings consisting of one X, m+n-1 Y's and m+n-1 Z's in which the X lies to the right of at least m Y's and at least m Z's.

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Author

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Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A380119 Triangle read by rows: T(n, k) is the number of walks of length 2*n on the N X N grid with unit steps in all four directions (NSWE) starting at (0, 0). k is the common value of the x- and the y-coordinate of the endpoint of the walk.

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Python