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
Links
- Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.
Crossrefs
Programs
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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
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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 *) -
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
Formula
Extensions
a(13)-a(25) from Mélika Tebni, Dec 06 2024
Comments