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A227665 Number of lattice paths from {n}^3 to {0}^3 using steps that decrement one component by 1 such that for each point (p_1,p_2,p_3) we have abs(p_{i}-p_{i+1}) <= 1.

%I A227665 #14 Oct 05 2021 16:29:01
%S A227665 1,6,44,320,2328,16936,123208,896328,6520712,47437640,345104904,
%T A227665 2510609608,18264477064,132872558664,966636864776,7032203170760,
%U A227665 51158695924872,372175277815624,2707544336559112,19697160911545032,143295215053933448,1042460827200624200
%N A227665 Number of lattice paths from {n}^3 to {0}^3 using steps that decrement one component by 1 such that for each point (p_1,p_2,p_3) we have abs(p_{i}-p_{i+1}) <= 1.
%H A227665 Alois P. Heinz, <a href="/A227665/b227665.txt">Table of n, a(n) for n = 0..1000</a>
%H A227665 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,2).
%F A227665 G.f.: (x-1)/(2*x^2+7*x-1).
%F A227665 a(n) = 7*a(n-1) + 2*a(n-2) for n>1, a(0)=1, a(2)=6.
%e A227665 a(1) = 3! = 3*2*1 = 6:
%e A227665             (0,1,1) - (0,0,1)
%e A227665           /         X         \
%e A227665   (1,1,1) - (1,0,1)   (0,1,0) - (0,0,0)
%e A227665           \         X         /
%e A227665             (1,1,0) - (1,0,0)
%p A227665 a:= n-> (<<0|1>, <2|7>>^n. <<1, 6>>)[1, 1]:
%p A227665 seq(a(n), n=0..25);
%Y A227665 Column k=3 of A227655.
%Y A227665 Cf. A000142.
%K A227665 nonn,easy
%O A227665 0,2
%A A227665 _Alois P. Heinz_, Jul 19 2013