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A358734
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Number of down-steps (1,-1) among all n-length nondecreasing Dyck paths with air pockets.
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2
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1, 0, 2, 3, 7, 15, 33, 72, 157, 341, 738, 1591, 3417, 7312, 15593, 33145, 70242, 148443, 312893, 657944, 1380437, 2890349, 6040258, 12600623, 26243057, 54572320, 113321233, 235002417, 486735682, 1006950771, 2080889013, 4295799336, 8859716317, 18255789317, 37584488418, 77315114215, 158923017417, 326432444848
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OFFSET
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2,3
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COMMENTS
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A Dyck path with air pockets is a nonempty lattice path in the first quadrant of Z^2 starting at the origin, ending on the x-axis, and consisting of up-steps (1,1) and down-steps (1,-k), k > 0, where two down-steps cannot be consecutive. It is then nondecreasing if the sequence of heights of its valleys is nondecreasing, i.e., the sequence of the minimal ordinates of the occurrences (1,-k)--(1,1), k>0, is nondecreasing from left to the right.
For all k>0, a(n-k) is the number of k-pyramids (i.e., k consecutive up-steps (1,1), then a down-step (1,-k)) among all (n-1)-length nondecreasing Dyck paths with air pockets.
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LINKS
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FORMULA
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G.f.: (x^2*(1 - x)*(x^5 - 2*x^3 + 5*x^2 - 4*x + 1))/((1 - 2*x)^2*(-x^2 - x + 1)).
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MATHEMATICA
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LinearRecurrence[{5, -7, 0, 4}, {1, 0, 2, 3, 7, 15, 33}, 50] (* Paolo Xausa, Jan 18 2024 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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