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A218274 Number of n-step paths from (0,0) to (1,0) where all diagonal, vertical and horizontal steps are allowed.

%I A218274 #33 Aug 29 2021 04:31:38
%S A218274 0,1,4,27,168,1140,7800,54845,390320,2815344,20494320,150442908,
%T A218274 1111782672,8264558016,61743361680,463306724595,3489942222624,
%U A218274 26378657835816,199991245341888,1520403553182800,11587257160313120,88506896001503616,677426230547667744
%N A218274 Number of n-step paths from (0,0) to (1,0) where all diagonal, vertical and horizontal steps are allowed.
%C A218274 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 A218274 Alois P. Heinz, <a href="/A218274/b218274.txt">Table of n, a(n) for n = 0..1000</a>
%e A218274 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 A218274 a:= proc(n) option remember; `if`(n<3, n^2,
%p A218274       ((9*n^4-9*n^3-8*n^2+4*n) *a(n-1)
%p A218274       +4*(n-1)*(27*n^3-84*n^2+80*n-21) *a(n-2)
%p A218274       +32*(3*n-1)*(n-1)*(n-2)^2 *a(n-3))/ (n*(n-1)*(n+1)*(3*n-4)))
%p A218274     end:
%p A218274 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 02 2012
%t A218274 a[n_] := a[n] = If[n<3, n^2,
%t A218274      ((9n^4-9n^3-8n^2+4n) a[n-1] +
%t A218274      4(n-1)(27n^3-84n^2+80n-21) a[n-2] +
%t A218274      32(3n-1)(n-1)(n-2)^2 a[n-3]) /
%t A218274      (n(n-1)(n+1)(3n-4))];
%t A218274 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Aug 29 2021, after _Alois P. Heinz_ *)
%o A218274 (Maxima)
%o A218274 a[0]:0$
%o A218274 a[1]:1$
%o A218274 a[2]:4$
%o A218274 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 A218274(n):=a[n]$
%o A218274 makelist(A218274(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */
%Y A218274 Cf. A094061.
%K A218274 nonn,easy
%O A218274 0,3
%A A218274 _Jon Perry_, Nov 01 2012
%E A218274 More terms from _Joerg Arndt_, Nov 02 2012