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 A212366 Number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 7). 2
 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 29, 38, 52, 76, 117, 184, 288, 442, 662, 972, 1414, 2063, 3047, 4572, 6952, 10645, 16303, 24857, 37672, 56821, 85541, 128948, 195103, 296548, 452501, 692053, 1058990, 1619311, 2473171, 3773889, 5757885, 8791090 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 FORMULA G.f. satisfies: A(x) = 1+A(x)*(x-x^7*(1-A(x))). a(n) = a(n-1) + Sum_{k=1..n-7} a(k)*a(n-7-k) if n>0; a(0) = 1. EXAMPLE a(0) = 1: the empty path. a(1) = 1: UD. a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD. a(9) = 4: UDUDUDUDUDUDUDUDUD, UDUUUUUUUUDDDDDDDD, UUUUUUUUDDDDDDDDUD, UUUUUUUUDUDDDDDDDD. MAPLE a:= proc(n) option remember;       `if`(n=0, 1, a(n-1) +add(a(k)*a(n-7-k), k=1..n-7))     end: seq(a(n), n=0..50); # second Maple program: a:= n-> coeff(series(RootOf(A=1+A*(x-x^7*(1-A)), A), x, n+1), x, n): seq(a(n), n=0..50); MATHEMATICA With[{k = 7}, CoefficientList[Series[(1 - x + x^k - Sqrt[(1 - x + x^k)^2 - 4*x^k]) / (2*x^k), {x, 0, 40}], x]] (* Vaclav Kotesovec, Sep 02 2014 *) CROSSREFS Column k=7 of A212363. Sequence in context: A225088 A175777 A098574 * A309838 A334251 A175776 Adjacent sequences:  A212363 A212364 A212365 * A212367 A212368 A212369 KEYWORD nonn AUTHOR Alois P. Heinz, May 10 2012 STATUS approved

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Last modified September 29 06:29 EDT 2020. Contains 337425 sequences. (Running on oeis4.)