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A307347 Number of 2n-move closed antelope paths on an unbounded chessboard from a given square to the same square.

%I A307347 #9 Apr 10 2019 02:51:56
%S A307347 1,8,168,5120,190120,7939008,357713664,17010543264,842994009000,
%T A307347 43192225007360,2275378947981568,122724475613935104,
%U A307347 6753785574641857024,378138077830110886400,21486835143540141873120,1236506847203439155401920,71934214120446285067176360
%N A307347 Number of 2n-move closed antelope paths on an unbounded chessboard from a given square to the same square.
%C A307347 Antelope is a leaper [3,4].
%H A307347 Vaclav Kotesovec, <a href="/A307347/b307347.txt">Table of n, a(n) for n = 0..552</a>
%F A307347 a(n) = the constant term in the expansion of (x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^(2*n).
%F A307347 Conjecture: a(n) ~ 64^n / (25*Pi*n).
%p A307347 b:= proc(n, x, y) option remember; `if`(max(x, y)>4*n or x+y>7*n, 0,
%p A307347       `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[4, 3],
%p A307347       [3, 4], [-4, 3], [-3, 4], [4, -3], [3, -4], [-4, -3], [-3, -4]])))
%p A307347     end:
%p A307347 a:= n-> b(2*n, 0$2):
%p A307347 seq(a(n), n=0..25);
%p A307347 # second Maple program:
%p A307347 poly := expand((x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do:
%t A307347 poly = Expand[(x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^2]; z = 1; Flatten[{1, Table[z = Expand[z*poly]; z[[1]], {n, 1, 15}]}]
%Y A307347 Cf. A094061, A253974, A254129, A254459.
%K A307347 nonn
%O A307347 0,2
%A A307347 _Vaclav Kotesovec_, Apr 03 2019