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A192446 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (3,0), (0,1), (0,3).

%I A192446 #36 Oct 06 2019 05:36:44
%S A192446 1,2,6,30,154,768,3906,20232,105750,556328,2943432,15646932,83500126,
%T A192446 447057380,2400249624,12918250836,69674241654,376489511460,
%U A192446 2037768450480,11045915485740,59955446568276,325821729044784,1772588671356204,9653187691115640,52617711157401186,287051310425050668
%N A192446 Number of lattice paths from (0,0) to (n,n) using steps (1,0), (3,0), (0,1), (0,3).
%C A192446 Diagonal of rational function 1/(1 - (x + y + x^3 + y^3)). - _Gheorghe Coserea_, Aug 06 2018
%H A192446 Alois P. Heinz, <a href="/A192446/b192446.txt">Table of n, a(n) for n = 0..1344</a> (first 304 terms from Gheorghe Coserea)
%F A192446 G.f.: sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3)  where s is a function satisfying 3*s^2*x^2-(1-2*s^2+2*s^3)*x-s*(s^3+2*s^2+s-1)=0. - _Mark van Hoeij_, Apr 17 2013
%F A192446 From _Gheorghe Coserea_, Aug 06 2018: (Start)
%F A192446 G.f. y=A(x) satisfies:
%F A192446 0 = (4*x^3 + 8*x^2 + 4*x - 1)^4*(108*x^3 - 108*x^2 + 36*x - 31)^2*y^8 + 4*(4*x^3 + 8*x^2 + 4*x - 1)^3*(36*x^3 + 36*x^2 - 4*x - 13)*(108*x^3 - 108*x^2 + 36*x - 31)*y^6 + 2*(4*x^3 + 8*x^2 + 4*x - 1)^2*(2160*x^6 + 4320*x^5 + 1872*x^4 - 1784*x^3 - 1576*x^2 + 472*x + 431)*y^4 + 4*(4*x^3 + 8*x^2 + 4*x - 1)*(112*x^6 + 448*x^5 + 688*x^4 + 456*x^3 + 96*x^2 + 40*x + 55)*y^2 + (4*x^3 + 12*x^2 + 12*x + 3)^2.
%F A192446 0 = (4*x^3 + 8*x^2 + 4*x - 1)*(108*x^3 - 108*x^2 + 36*x - 31)*(270*x^4 + 180*x^3 + 144*x^2 - 225*x - 59)*y''' + (1283040*x^9 + 1924560*x^8 + 1080864*x^7 - 1425816*x^6 - 2135376*x^5 + 33048*x^4 + 702468*x^3 + 134520*x^2 + 43892*x + 30575)*y'' + 30*(111780*x^8 + 149040*x^7 + 120960*x^6 - 122094*x^5 - 172206*x^4 - 6012*x^3 + 36615*x^2 + 10298*x - 1541)*y' + 60*(29160*x^7 + 34020*x^6 + 36288*x^5 - 43092*x^4 - 45882*x^3 - 6462*x^2 + 1890*x + 913)*y.
%F A192446 (End)
%p A192446 REL := 3*s^2*x^2-(1-2*s^2+2*s^3)*x-s*(s^3+2*s^2+s-1);
%p A192446 ogf := sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3);
%p A192446 series(eval(ogf, s=RootOf(REL,s)),x=0,30);  # _Mark van Hoeij_, Apr 17 2013
%p A192446 # second Maple program:
%p A192446 b:= proc(x, y) option remember; `if`(y=0, 1, add((p->
%p A192446       `if`(p[1]<0, 0, b(p[1], p[2])))(sort([x, y]-h)),
%p A192446         h=[[1, 0], [0, 1], [3, 0], [0, 3]]))
%p A192446     end:
%p A192446 a:= n-> b(n$2):
%p A192446 seq(a(n), n=0..30);  # _Alois P. Heinz_, Dec 28 2018
%t A192446 a[0, 0] = 1; a[n_, k_] /; n >= 0 && k >= 0 := a[n, k] = a[n, k-1] + a[n, k-3] + a[n-1, k] + a[n-3, k]; a[_, _] = 0;
%t A192446 a[n_] := a[n, n];
%t A192446 a /@ Range[0, 30] (* _Jean-François Alcover_, Oct 06 2019 *)
%o A192446 (PARI) /* same as in A092566 but use */
%o A192446 steps=[[1,0], [3,0], [0,1], [0,3]];
%o A192446 /* _Joerg Arndt_, Jun 30 2011 */
%o A192446 (PARI)
%o A192446 seq(N) = {
%o A192446   my(x='x + O('x^N), d=16*x^6 + 16*x^5 + 16*x^4 - 8*x^3 - 4*x^2 + 1,
%o A192446      s=serreverse((1 - 2*x^2 + 2*x^3 - sqrt(d))/(6*x^2)));
%o A192446   Vec(sqrt((8*s^2*(x*(1-8*s^2-4*s^3)-2*s^4-4*s^3-8*s^2+2*s+3)/3-1)/(4*x^3+8*x^2+4*x-1))/(1-4*s^3));
%o A192446 };
%o A192446 seq(26) \\ _Gheorghe Coserea_, Aug 06 2018
%Y A192446 Cf. A001850, A026641, A036355, A137644, A192364, A192365, A192369, A192368.
%K A192446 nonn
%O A192446 0,2
%A A192446 _Joerg Arndt_, Jul 01 2011