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

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

0,2

Comments

See Dershowitz (2017) for precise definition.

Links

Table of n, a(n) for n=0..25.

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

Formula

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)

Maple

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

Mathematica

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 *)

Prog

(Python)
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

Crossrefs

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

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

Sequence in context: A150430 A363434 A150431 * A150432 A150433 A197554

Adjacent sequences: A302182 A302183 A302184 * A302186 A302187 A302188

Keyword

nonn,walk

Author

N. J. A. Sloane, Apr 09 2018

Extensions

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

Status

approved