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%I #6 Oct 06 2015 18:04:37
%S 1,1,1,6,3,1,44,16,5,1,344,116,30,7,1,2856,928,224,48,9,1,24816,7856,
%T 1840,376,70,11,1,223016,69264,15912,3184,580,96,13,1,2056256,629472,
%U 142592,28176,5080,844,126,15,1,19344472,5855472,1312360,256992,46072
%N Triangle read by rows: T(n,k) is 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 have k pyramids of the first kind (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).
%C Also 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 have k pyramids of the second kind (a pyramid of the second kind is a sequence U^pd^(2p) for some positive integer p, starting at the x-axis). Row sums yield A027307. Column 0 yields A108452. Number of pyramids of the first kind in all paths from (0,0) to (3n,0) is given by A108453.
%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-tz-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).
%e T(2,1)=3 because we have (ud)Udd, (uudd) and Udd(ud), the pyramids of the first kind being shown between parentheses.
%e Triangle begins:
%e 1;
%e 1,1;
%e 6,3,1;
%e 44,16,5,1;
%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-t*z-z*(1-z)*A*(1+A)): Gser:=simplify(series(G,z=0,13)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 9 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form
%Y Cf. A027307, A108452, A108453, A108445.
%K nonn,tabl
%O 0,4
%A _Emeric Deutsch_, Jun 11 2005
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