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 A212385 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 5). 7
 1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 943, 1873, 3914, 9101, 23298, 61915, 162283, 409888, 996456, 2360486, 5555333, 13244114, 32357022, 80958851, 205389082, 522000262, 1317987172, 3297123652, 8190326857, 20302864970, 50482613327, 126318440989 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Lengths of descents are unrestricted. The radius of convergence of g.f. A(x) is r = 5*(1-2*s+s^2)/(s*(5*s-4)) = 0.3804593157188..., where s = A(r) is described below. - Vaclav Kotesovec, Mar 20 2014 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Vaclav Kotesovec, Recurrence (of order 10) Vaclav Kotesovec, Asymptotic of subsequences of A212382 FORMULA G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^5). Representation in terms of special values of generalized hypergeometric function of type 12F11: a(n) = hypergeom([1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -(1/6)*n, -(1/6)*n+5/6, -(1/6)*n+2/3, -(1/6)*n+1/2, -(1/6)*n+1/3, 1/6-(1/6)*n], [1/6, 1/3, 1/3, 1/2, 1/2, 2/3, 2/3, 5/6, 5/6, 1, 7/6], 7^7/6^6), n>=0. - Karol A. Penson, Jun 21 2013 a(n) ~ s^(n+3/2) * (5*s-4)^(n+2) / (2 * sqrt(Pi) * sqrt(3*s-2) * n^(3/2) * 5^(n+5/2) * (s-1)^(2*n+9/2)), where s = 1.87696911628429... is the root of the equation 2869 - 29970*s + 138225*s^2 - 373000*s^3 + 655625*s^4 - 787500*s^5 + 656250*s^6 - 375000*s^7 + 140625*s^8 - 31250*s^9 + 3125*s^10 = 0. - Vaclav Kotesovec, Mar 20 2014 a(n) = Sum_{k=0..n}(binomial(4*k-3*n-1,n-k)*binomial(n+1,5*k-4*n))/(n+1). - Vladimir Kruchinin, Mar 05 2016 EXAMPLE a(0) = 1: the empty path. a(1) = 1: UD. a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD. a(7) = 8: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDDDDDUDD, UUUUUUDDDDUDDD, UUUUUUDDDUDDDD, UUUUUUDDUDDDDD, UUUUUUDUDDDDDD. MAPLE b:= proc(x, y, u) option remember; `if`(x<0 or y b(n\$2, true): seq(a(n), n=0..40); # second Maple program a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^5), A), x, n+1), x, n): seq(a(n), n=0..40); MATHEMATICA b[x_, y_, u_] := b[x, y, u] = If[x<0 || y

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Last modified March 3 11:44 EST 2024. Contains 370511 sequences. (Running on oeis4.)