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A110220
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Triangle read by rows: T(n,k) (0<=k<=floor(n/2)) is the number of lattice paths from (0,0) to (2n,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0), never going below the x-axis (i.e. Schroeder paths) and having k UH's.
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1
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1, 2, 5, 1, 15, 7, 51, 37, 2, 188, 181, 25, 731, 866, 204, 5, 2950, 4124, 1393, 91, 12235, 19657, 8672, 1008, 14, 51822, 93937, 51147, 8856, 336, 223191, 450220, 291470, 68085, 4710, 42, 974427, 2163910, 1622665, 480535, 50655, 1254, 4302645
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Row n has 1+floor(n/2) terms. Row sums yield the large Schroeder numbers (A006318). T(n,0)=A007317(n+1). sum(k*T(n,k),k=0..floor(n/2))=A026002(n-1) for n>=2. T(2n,n)=Cat(n) (the n-th Catalan number; A000108).
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FORMULA
| G.f.=[1-z-sqrt(1-6z+5z^2-4tz^2)]/[2z(1-z+tz)].
T(n,k)=binomial(n+1,k)*Sum(binomial(n+1-k,k+j+1)*binomial(2j+2k,j), j=0..n-2k)/(n+1). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 28 2007
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EXAMPLE
| T(3,1)=7 because we have HUHD, UHDH, UHDUD, UHHD, UHUDD, UUHDD and UDUHD.
Triangle starts:
1;
2;
5,1;
15,7;
51,37,2;
188,181,25;
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MAPLE
| G:=(1-z-sqrt(1-6*z+5*z^2-4*z^2*t))/2/z/(1-z+t*z): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 12 do seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A006318, A007317, A026002, A000108.
Sequence in context: A177267 A178978 A101895 * A119518 A186756 A184940
Adjacent sequences: A110217 A110218 A110219 * A110221 A110222 A110223
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jul 16 2005
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