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Number of Deutsch paths with peaks at odd height.

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`%I #17 Mar 06 2022 08:14:42
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`%S 1,0,1,0,2,2,6,11,26,56,129,294,684,1599,3774,8961,21411,51421,124081,
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`%T 300667,731337,1785010,4370431,10731270,26419202,65198847,161262046,
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`%U 399692001,992559011,2469265633,6153306125,15357906136,38388056063,96086525311,240821963528
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`%N Number of Deutsch paths with peaks at odd height.
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`%C a(n) is the number of closed Deutsch paths of n steps with all peaks at odd height. A Deutsch path is a lattice path of up-steps (1,1) and down-steps (1,-j), j>=1, starting at the origin that never goes below the x-axis, and it is closed if it ends on the x-axis.
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`%C A166300 counts closed Deutsch paths with all peaks at even height.
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`%H Helmut Prodinger, <a href="https://arxiv.org/abs/2003.01918">Deutsch paths and their enumeration</a>, preprint, arXiv:2003.01918 [math.CO], 2020.
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`%F With F = 1 + x^2 + 2*x^4 + 2*x^5+ ... the g.f. for Deutsch paths with all peaks at odd height and G = 1 + x^3 + x^4 + 2*x^5+ ... the g.f. for Deutsch paths with all peaks at even height, a count based on the decomposition of paths according to the size j of the first down-step (1,-j) that returns the path to ground level yields the pair of simultaneous equations
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`%F F = 1 + (x^2*F*G + x^3*(F-1)*F*G)/(1 - x^2*F*G),
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`%F G = 1 + (x^2*(F-1)*G + x^3*F*G^2)/(1 - x^2*F*G).
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`%F G.f.: (1 + x + x^2 - sqrt[(1 - 3*x + x^2)*(1 + x + x^2)])/(2*x*(1 + x)).
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`%F D-finite with recurrence (n+1)*a(n) +(-n+2)*a(n-1) +3*(-n+1)*a(n-2) +3*(-n+3)*a(n-3) +(-n+2)*a(n-4) +(n-5)*a(n-5)=0. - _R. J. Mathar_, Mar 06 2022
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`%e a(5) = 2 counts UUU12, UUU21, where U denotes an up-step and a down-step is denoted by its length, and a(6) = 6 counts UUUUU5, UUU1U3, UUU111, UUU3U1, U1UUU3, U1U1U1.
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`%t CoefficientList[Series[(1 + x + x^2 - Sqrt[(1 - 3 x + x^2) (1 + x + x^2)])/(2 x + 2 x^2), {x, 0, 20}], x]
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`%Y Cf. A166300.
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`%K nonn,easy
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`%O 0,5
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`%A _David Callan_, Dec 14 2021
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