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%I #12 Jul 26 2022 15:26:06
%S 1,0,4,32,252,2112,18484,166976,1545548,14583808,139774180,1356966240,
%T 13316740764,131890671680,1316627340564,13234192747648,
%U 133829733962732,1360586260341248,13898403178004420,142578916276009632
%N Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and having no pyramids (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).
%C Column 0 of A108445.
%H Vaclav Kotesovec, <a href="/A108449/b108449.txt">Table of n, a(n) for n = 0..100</a>
%H Emeric Deutsch, <a href="http://www.jstor.org/stable/2589192">Problem 10658: Another Type of Lattice Path</a>, American Math. Monthly, 107, 2000, 368-370.
%F G.f.=(1-z)/[1+z-z(1-z)A(1+A)], where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
%F G.f. y(x) satisfies: -1 + 3*x - 3*x^2 + x^3 + y + 3*x*y - 9*x^2*y + 5*x^3*y - 5*x*y^2 - x^2*y^2 + 5*x^3*y^2 + x^4*y^2 - x*y^3 + 9*x^2*y^3 - 3*x^3*y^3 + 3*x^4*y^3 = 0. - _Vaclav Kotesovec_, Mar 18 2014
%F a(n) ~ (11+5*sqrt(5))^n * sqrt(1738885 + 811683*sqrt(5)) / (961*sqrt(5*Pi) *n^(3/2)*2^(n+3/2)). - _Vaclav Kotesovec_, Mar 18 2014
%F Conjecture D-finite with recurrence +n*(2*n+1)*(431*n-2895)*a(n) +2*(-9395*n^3+68622*n^2-64084*n+26109)*a(n-1) +2*(59288*n^3-508196*n^2+1044822*n-574587)*a(n-2) +2*(-94965*n^3+1070605*n^2-3607435*n+3485484)*a(n-3) +4*(29036*n^3-351474*n^2+1402336*n-1970505)*a(n-4) +6*(-6703*n^3+63052*n^2-99178*n-237177)*a(n-5) +6*(1012*n^3-14914*n^2+74580*n-127341)*a(n-6) +6*(1127*n^3-21429*n^2+135199*n-282762
%F )*a(n-7) +9*(29*n-165)*(2*n-15)*(n-8)*a(n-8)=0. - _R. J. Mathar_, Jul 26 2022
%e a(2)=4 because the paths uUddd, Ududd, UdUddd and Uuddd have no pyramids.
%p A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=(1-z)/(1+z-z*(1-z)*A*(1+A)): gser:=series(g,z=0,24): 1,seq(coeff(gser,z^n),n=1..21);
%o (PARI) {a(n)=local(y=1); for(i=1, n, y = -(-1 + 3*x - 3*x^2 + x^3 + 3*x*y - 9*x^2*y + 5*x^3*y - 5*x*y^2 - x^2*y^2 + 5*x^3*y^2 + x^4*y^2 - x*y^3 + 9*x^2*y^3 - 3*x^3*y^3 + 3*x^4*y^3) + (O(x^n))^4); polcoeff(y, n)}
%o for(n=0, 20, print1(a(n), ", ")) \\ _Vaclav Kotesovec_, Mar 18 2014
%Y Cf. A027307, A108445.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Jun 11 2005
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