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A127620 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 6 with the steps (1,0), (0, 1), (2,0) and (0,2).

%I A127620 #14 Apr 03 2019 10:42:21
%S A127620 1,1,5,22,117,654,3843,22882,137443,827998,4995443,30155494,182083275,
%T A127620 1099560942,6640309323,40101959542,242184540139,1462610652718,
%U A127620 8833070227499,53345145429670,322164911643723,1945636121710110
%N A127620 Number of walks from (0,0) to (n,n) in the region 0 <= x-y <= 6 with the steps (1,0), (0, 1), (2,0) and (0,2).
%H A127620 Arvind Ayyer and Doron Zeilberger, <a href="https://arxiv.org/abs/math/0610734">The Number of [Old-Time] Basketball games with Final Score n:n where the Home Team was never losing but also never ahead by more than w Points</a>, arXiv:math/0610734 [math.CO], 2006-2007.
%H A127620 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6, 5, -24, -28, -6, 8).
%F A127620 G.f.: (1 - 5x - 6x^2 + 11x^3 + 12x^4 - 4x^5)/(1 - 6x - 5x^2 + 24x^3 + 28x^4 + 6x^5 - 8x^6). [corrected by _Jean-François Alcover_, Apr 02 2019]
%e A127620 a(2)=5 because we can reach (2,2) in the following ways:
%e A127620 (0,0),(1,0),(1,1),(2,1),(2,2)
%e A127620 (0,0),(2,0),(2,2)
%e A127620 (0,0),(1,0),(2,0),(2,2)
%e A127620 (0,0),(2,0),(2,1),(2,2)
%e A127620 (0,0),(1,0),(2,0),(2,1),(2,2)
%t A127620 b[n_, k_] := Boole[n >= 0 && k >= 0 && 0 <= n-k <= 6];
%t A127620 T[0, 0] = T[1, 1] = 1; T[n_, k_] /; b[n, k] == 1 := T[n, k] = b[n-1, k]* T[n-1, k] + b[n-2, k]*T[n-2, k] + b[n, k-1]*T[n, k-1] + b[n, k-2]*T[n, k-2]; T[_, _] = 0;
%t A127620 a[n_] := T[n, n];
%t A127620 Table[a[n], {n, 0, 21}]
%t A127620 (* or: *)
%t A127620 LinearRecurrence[{6, 5, -24, -28, -6, 8}, {1, 1, 5, 22, 117, 654}, 22] (* _Jean-François Alcover_, Apr 02 2019 *)
%Y A127620 Cf. A000108, A046717, A122951, A127617, A127618, A127619.
%K A127620 nonn,easy,walk
%O A127620 0,3
%A A127620 _Arvind Ayyer_, Jan 20 2007