A005558 -id:A005558 - cp's OEIS frontend

A289352 Irregular triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with floor((n+2)/2) up movements in odd numbered positions and k returns to the x axis.

1, 0, 1, 1, 2, 1, 2, 3, 6, 8, 6, 10, 15, 15, 10, 50, 60, 45, 20, 105, 140, 126, 84, 35, 490, 560, 420, 224, 70, 1176, 1470, 1260, 840, 420, 126, 5292, 5880, 4410, 2520, 1050, 252, 13860, 16632, 13860, 9240, 4950, 1980, 462, 60984, 66528, 49896, 29568, 13860, 4752, 924

Offset: 1

Roger Ford, Jul 03 2017

			n\k  1    2    3    4    5
1    1
2    0    1
3    1    2
4    1    2    3
5    6    8    6
6   10   15   15   10
7   50   60   45   20
8  105  140  126   84   35
9  490  560  420  224   70
T(4,3)=3: (U = up in odd position, u = up in even position, d = down, _ = return to x axis, floor ((n+2)/2) = 3 up movements in odd position)  Ud_Ud_Uudd_, Uudd_Ud_Ud_, Ud_Uudd_Ud_.

Cf. A001263, A001405, A005558.

T(1,1)=1, T(2,1)=0, T(2,2)=1, For n >= 3, T(n,k) = (1/floor((n-1)/2))*C(n-1,floor((n-3)/2))*C(n-1-k,floor((n-3)/2))*k (conjectured).
Row sums of T(n,k) = A005558(a(n-1)).
T(n,1) = A001263(T(n-1,floor(n/2)).
T(n,floor((n+2)/2)) = A001405(a(n-1)).

A297672 Array with four columns read by rows: T(n,k) = number of n step walks in the first octant on a square lattice with last step being right (k=1), left (k=2), up (k=3) or down (k=4).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 1, 0, 3, 1, 1, 1, 6, 6, 6, 2, 20, 10, 10, 10, 50, 50, 50, 25, 175, 105, 105, 105, 490, 490, 490, 294, 1764, 1176, 1176, 1176, 5292, 5292, 5292, 3528, 19404, 13860, 13860

Offset: 1

Roger Ford, Jan 02 2018

Sum_{k=1..4} T(n,k) = A005558(n).

For n >= 1, the ratio of the numbers of right or up last steps to left or down last steps is floor((n+2)/2): floor(n/2). - Roger Ford, Oct 28 2019

			k=    1    2    3    4    total
N   right left up   down  walks
1     1    0    0    0    =1
2     1    1    1    0    =3
3     3    1    1    1    =6
4     6    6    6    2    =20
There are 6 walks of 4 steps in the octant with the last step right. T(4,1)=6 RRRR, RRLR, RLRR, RUDR, RURR, RRUR.

Cf. A005558, A001263.

n=1: T(1,1)=1, T(1,2)=0, T(1,3)=0, T(1,4)=0;
n>1: T(n,1) = C(n,floor(n/2))*C(n-1,floor((n-1)/2)) - C(n,floor((n-1)/2))*C(n-1,floor((n-2)/2));
T(n,2) = T(n,3) = C(n-1,floor(n/2)-1)*C(n,floor(n/2)-1)/floor(n/2);
n odd:  T(n,4) = T(n,2);
n even: T(n,4) = T(n,2)*((n/2-1)/(n/2+1));
For n > 1, T(n,2) = T(n,3) = A001263(n,floor(n/2)).

A298122 a(n) is the number of n nonintersecting arches above the x-axis that start and/or end with an arch length equal to one and have floor((n+2)/2) arches starting in odd numbered positions.

Original entry on oeis.org

1, 1, 2, 5, 11, 34, 90, 300, 875, 3038, 9408, 33516, 108108, 392040, 1302444, 4785066, 16256955, 60324550, 208579800, 780088452, 2735682092, 10296854984, 36532677272, 138231751840, 495241833996, 1882201158264, 6799413051200, 25939319270000, 94374970110000

Offset: 1

Roger Ford, Jan 12 2018

nonn

			Example: For n = 4 the a(4) = 5 solutions are as follows. (The numbers under the arches represent arches starting in an odd-numbered position on the x-axis.)
    /\                                /\
   //\\      /\            /\        //\\        /\
  ///\\\/\, //\\ /\ /\, /\//\\/\, /\///\\\, /\/\//\\.
  1 3   7   1    5  7   1 3   7   1 3 5     1 3 5

Cf. A001263, A005558.

a(1) = a(2) = 1, a(3) = 2; for n > 3, a(n) = 2*(C(n-1, floor((n-1)/2))*C(n-2, floor((n-2)/2)) - (C(n-1, floor((n-2)/2))*C(n-2, floor((n-3)/2)))) - (C(n-3, floor((n-1)/2))*C(n-2, floor((n-1)/2))/(floor((n-1)/2)+1)).

A302185 Number of 3D n-step walks of type acc.

Original entry on oeis.org

1, 2, 7, 24, 98, 400, 1785, 7980, 37674, 178164, 874146, 4294752, 21667932, 109436184, 563910633, 2908233900, 15235550330, 79870553620, 424021948350, 2252356700880, 12088746573540, 64913104882080, 351594254659830, 1905139854213960, 10399223643879420, 56783986550235000

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A000918, A001405, A005558, A005566, A135394, A302182.

b:= n-> binomial(n, floor(n/2))*binomial(n+1, floor((n+1)/2)):
C:= n-> binomial(2*n, n)/(n+1):
a:= n-> add(binomial(n, 2*k)*C(k)*b(n-2*k), k=0..n/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 06 2024
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [1, 2, 7, 24][n+1],
      (8*(14*n^4+85*n^3+190*n^2+188*n+63)*a(n-1)+4*(n-1)*
      (80*n^4+418*n^3+676*n^2+269*n-108)*a(n-2)-96*(n-1)*(n-2)*
      (10*n^2+31*n+27)*a(n-3)-144*(n-1)*(n-2)*(n-3)*(8*n^2+33*n+36)*
       a(n-4))/((n+4)*(n+3)*(n+2)*(8*n^2+17*n+11)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 06 2024

b[n_] := Binomial[n, Floor[n/2]]*Binomial[n+1, Floor[(n+1)/2]];
c[n_] := Binomial[2*n, n]/(n+1);
a[n_] := Sum[Binomial[n, 2*k]*c[k]*b[n - 2*k], {k, 0, n/2}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jan 28 2025, after Alois P. Heinz *)

from math import comb as binomial
def C(n): return (binomial(2*n, n)//(n+1)) # Catalan numbers
def a(n):
    return sum(binomial(n, k)*C((k+1)//2)*C(k//2)*(2*(k//2)+1)*binomial(n-k, (n-k)//2) for k in range(n+1))
print([a(n) for n in range(26)]) # Mélika Tebni, Dec 06 2024

From Mélika Tebni, Dec 06 2024: (Start)
E.g.f.: (BesselI(0, 2*x) + BesselI(1, 2*x))^2*BesselI(1, 2*x) / x.
a(n) = Sum_{k=0..n} binomial(n, k)*A005558(k)*A001405(n-k).
a(2*n+1) = 2*A302182(2*n+1) = A135394(n) / (n+1).
For n > 0, a(A000918(n)) is odd. (End)

a(13)-a(25) from Mélika Tebni, Dec 06 2024

cp's OEIS Frontend

A289352 Irregular triangle read by rows: T(n,k) = the number of Dyck paths of semilength n with floor((n+2)/2) up movements in odd numbered positions and k returns to the x axis.

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A297672 Array with four columns read by rows: T(n,k) = number of n step walks in the first octant on a square lattice with last step being right (k=1), left (k=2), up (k=3) or down (k=4).

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A298122 a(n) is the number of n nonintersecting arches above the x-axis that start and/or end with an arch length equal to one and have floor((n+2)/2) arches starting in odd numbered positions.

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A302185 Number of 3D n-step walks of type acc.

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