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A108747
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Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n and having k returns to the x-axis.
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2
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2, 2, 4, 4, 8, 8, 10, 20, 24, 16, 28, 56, 72, 64, 32, 84, 168, 224, 224, 160, 64, 264, 528, 720, 768, 640, 384, 128, 858, 1716, 2376, 2640, 2400, 1728, 896, 256, 2860, 5720, 8008, 9152, 8800, 7040, 4480, 2048, 512, 9724, 19448, 27456, 32032, 32032, 27456, 19712, 11264, 4608, 1024
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OFFSET
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1,1
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COMMENTS
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A Grand Dyck path of semilength n is a path in the half-plane x >= 0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1).
Triangle T(n,k), 1 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] DELTA [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jun 29 2005
T(n,k) is also equal to the number of grand Dyck paths of semilength n having k symmetric vertices. A symmetric vertex is a vertex in the first half of the path (not including the midpoint) that is a mirror image of a vertex in the second half, when with respect to the reflection along the vertical line through the midpoint of the path. - Sergi Elizalde, Feb 12 2021
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LINKS
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FORMULA
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T(n,n) = 2^n.
T(n,k) = k * 2^k * binomial(2*n-k,n)/(2*n-k) (1 <= k <= n).
G.f.: 1/(1-2*t*z*C), where C = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
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EXAMPLE
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T(2,2)=4 because we have u(d)u(d), u(d)d(u), d(u)d(u) and d(u)u(d) (return steps to x-axis shown between parentheses).
Triangle begins:
2;
2, 4;
4, 8, 8;
10, 20, 24, 16;
28, 56, 72, 64, 32;
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MAPLE
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T:= (n, k)-> 2^k*k*binomial(2*n-k, n)/(2*n-k): for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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MATHEMATICA
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nn=10; c=(1-(1-4x)^(1/2))/(2x); f[list_]:=Select[list, #>0&]; Map[f, Drop[CoefficientList[Series[1/(1-2y x c), {x, 0, nn}], {x, y}], 1]]//Flatten (* Geoffrey Critzer, Jan 30 2012 *)
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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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