A218274 - OEIS

%I #33 Aug 29 2021 04:31:38

%S 0,1,4,27,168,1140,7800,54845,390320,2815344,20494320,150442908,

%T 1111782672,8264558016,61743361680,463306724595,3489942222624,

%U 26378657835816,199991245341888,1520403553182800,11587257160313120,88506896001503616,677426230547667744

%N Number of n-step paths from (0,0) to (1,0) where all diagonal, vertical and horizontal steps are allowed.

%C Equivalent to which linear combinations of (-1,-1), (-1,0), (-1,1), (0,1), (0,-1), (1,1), (1,0), (1,-1) equal (1,0).

%H Alois P. Heinz, <a href="/A218274/b218274.txt">Table of n, a(n) for n = 0..1000</a>

%e a(2) = 4 because we have [0,1]+[1,-1], [1,1]+[0,-1] and the y-negatives [0,-1]+[1,1], [1,-1]+[0,1].

%p a:= proc(n) option remember; `if`(n<3, n^2,

%p       ((9*n^4-9*n^3-8*n^2+4*n) *a(n-1)

%p       +4*(n-1)*(27*n^3-84*n^2+80*n-21) *a(n-2)

%p       +32*(3*n-1)*(n-1)*(n-2)^2 *a(n-3))/ (n*(n-1)*(n+1)*(3*n-4)))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 02 2012

%t a[n_] := a[n] = If[n<3, n^2,

%t      ((9n^4-9n^3-8n^2+4n) a[n-1] +

%t      4(n-1)(27n^3-84n^2+80n-21) a[n-2] +

%t      32(3n-1)(n-1)(n-2)^2 a[n-3]) /

%t      (n(n-1)(n+1)(3n-4))];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 29 2021, after _Alois P. Heinz_ *)

%o (Maxima)

%o a[0]:0$

%o a[1]:1$

%o a[2]:4$

%o a[n]:= ((9*n^4-9*n^3-8*n^2+4*n)*a[n-1]+4*(n-1)*(27*n^3-84*n^2+80*n-21)*a[n-2]+32*(3*n-1)*(n-1)*(n-2)^2 *a[n-3])/(n*(n-1)*(n+1)*(3*n-4))$

%o A218274(n):=a[n]$

%o makelist(A218274(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */

%Y Cf. A094061.

%K nonn,easy

%O 0,3

%A _Jon Perry_, Nov 01 2012

%E More terms from _Joerg Arndt_, Nov 02 2012