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A109157 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 having sum of the heights of its pyramids equal to k (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis; p is the height of the pyramid). 0
1, 0, 1, 1, 4, 0, 2, 2, 2, 32, 8, 8, 4, 5, 5, 4, 252, 64, 84, 24, 28, 12, 14, 12, 8, 2112, 520, 680, 240, 232, 88, 76, 37, 37, 28, 16, 18484, 4480, 5804, 1992, 2012, 776, 656, 264, 206, 106, 94, 64, 32, 166976, 40008, 51592, 17440, 17400, 6776, 5680, 2392, 1768 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Row n has 2n+1 terms. Row sums yield A027307. Column 0 yields A108449.
LINKS
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
FORMULA
G.f. = (1-z)(1-tz)(1-t^2*z)/[1-2tz-2t^2*z+z+3t^3*z^2-t^3*z^3-z(1-z)(1-t^2*z)(1-tz)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).
EXAMPLE
T(2,3)=2 because we have udUdd and Uddud.
Triangle begins:
1;
0,1,1;
4,0,2,2,2;
32,8,8,4,5,5,4;
252,64,84,24,28,12,14,12,8;
MAPLE
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)*(-1+t^2*z)/(z*(-1+z)*(-1+t^2*z)*(-1+t*z)*A*(1+A)+1-2*t*z-2*t^2*z+z+3*t^3*z^2-t^3*z^3): Gser:=simplify(series(G, z=0, 10)): P[0]:=1: for n from 1 to 7 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 7 do seq(coeff(t*P[n], t^k), k=1..2*n+1) od; # yields sequence in triangular form
CROSSREFS
Sequence in context: A320479 A218769 A180850 * A226955 A123314 A058997
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 20 2005
STATUS
approved

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Last modified May 23 11:42 EDT 2024. Contains 372760 sequences. (Running on oeis4.)