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 A243965 Number of Dyck paths of semilength n such that both consecutive patterns of Dyck paths of semilength 2 occur at least once. 3
 0, 0, 0, 0, 2, 10, 44, 179, 702, 2701, 10278, 38866, 146450, 550817, 2070116, 7779655, 29248932, 110047905, 414446256, 1562538171, 5898049688, 22290789562, 84351810044, 319609669957, 1212552963576, 4606078246284, 17518748817596, 66712192842068, 254346235738120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS The consecutive patterns 1010, 1100 are counted. Here 1=Up=(1,1), 0=Down=(1,-1). LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 Vaclav Kotesovec, Recurrence (of order 10) FORMULA a(n) ~ 4^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 18 2014 EXAMPLE a(4) = 2: 10101100, 11001010. a(5) = 10: 1010101100, 1010110010, 1010111000, 1011001010, 1100101010, 1100110100, 1101001100, 1101011000, 1110001010, 1110010100. Here 1=Up=(1,1), 0=Down=(1,-1). MAPLE b:= proc(x, y, t, s) option remember; `if`(y<0 or y>x, 0, `if`(x=0, `if`(s={}, 1, 0), `if`(nops(s)>x, 0, add( b(x-1, y-1+2*j, irem(2*t+j, 8), s minus {2*t+j}), j=0..1)))) end: a:= n-> b(2*n, 0, 0, {10, 12}): seq(a(n), n=0..30); MATHEMATICA b[x_, y_, t_, s_] := b[x, y, t, s] = If[y<0 || y>x, 0, If[x == 0, If[s == {}, 1, 0], If[Length[s] > x, 0, Sum[b[x - 1, y - 1 + 2 j, Mod[2t + j, 8], s ~Complement~ {2t + j}], {j, 0, 1}]]]]; a[n_] := b[2n, 0, 0, {10, 12}]; a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *) CROSSREFS Cf. A014486, A063171, A243820, A243966. Sequence in context: A025590 A122932 A080069 * A218780 A068551 A099919 Adjacent sequences: A243962 A243963 A243964 * A243966 A243967 A243968 KEYWORD nonn AUTHOR Alois P. Heinz, Jun 16 2014 STATUS approved

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Last modified August 14 21:44 EDT 2024. Contains 375167 sequences. (Running on oeis4.)