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A121469
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Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n having k 1-cell columns (0<=k<=n).
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2
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1, 0, 1, 1, 0, 1, 1, 3, 0, 1, 3, 4, 5, 0, 1, 6, 13, 7, 7, 0, 1, 14, 28, 27, 10, 9, 0, 1, 31, 70, 62, 45, 13, 11, 0, 1, 70, 164, 171, 108, 67, 16, 13, 0, 1, 157, 392, 429, 325, 166, 93, 19, 15, 0, 1, 353, 926, 1101, 862, 540, 236, 123, 22, 17, 0, 1, 793, 2189, 2766, 2355, 1499, 824
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OFFSET
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0,8
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COMMENTS
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Also number of nondecreasing Dyck paths of semilength n and such that there are k ascents of length 1. A nondecreasing Dyck path is a Dyck path for which the sequence of the altitudes of the valleys is nondecreasing. Example: T(4,2)=5 because we have (U)D(U)DUUDD, (U)DUUDD(U)D, (U)DUUD(U)DD, UUDD(U)D(U)D and UUD(U)D(U)DD, where U=(1,1) and D=(1,-1); the ascents of length one are shown between parentheses (also the Dyck path UUDUDDUD has two ascents but it is not nondecreasing because the valleys have altitudes 1 and 0). Row sums are the odd-subscripted Fibonacci numbers (A001519). T(n,0)=A006356(n-3). Sum(k*T(n,k),k=0..n)=A094864(n-1).
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LINKS
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FORMULA
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G.f.: G(t,z)=(1-2z)/[1-(t+2)z+(2t-1)z^2-(t-1)z^3)].
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EXAMPLE
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T(3,1)=3 because we have the three directed column-convex polyominoes: [(0,2),(0,1)], [(0,2),(1,2)] and [(0,1),(0,2)] (here the j-th pair within the square brackets gives the lower and upper levels of the j-th column of that particular polyomino).
Triangle starts:
1;
0,1;
1,0,1;
1,3,0,1;
3,4,5,0,1;
6,13,7,7,0,1;
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MAPLE
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G:=(1-2*z)/(1-(t+2)*z+(2*t-1)*z^2-(t-1)*z^3): Gser:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 13 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 13 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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