A243412 Number of Dyck paths of semilength n avoiding the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)).

1, 1, 2, 5, 13, 37, 112, 352, 1136, 3742, 12529, 42513, 145868, 505234, 1764157, 6203370, 21947490, 78072209, 279062937, 1001803617, 3610366030, 13057141261, 47373444827, 172381857939, 628944880851, 2300410562946, 8433110899963, 30980398420830, 114034887644860

Offset: 0

Author

Alois P. Heinz, Jun 04 2014

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=0 of A243366.

Column k=45 of A243753.

Formula

Recurrence: (n+1)*(n+2)*(817*n^7 - 24387*n^6 + 285094*n^5 - 1647261*n^4 + 4787137*n^3 - 5628540*n^2 - 1552284*n + 6122952)*a(n) = (n+1)*(1634*n^8 - 47957*n^7 + 542786*n^6 - 2900786*n^5 + 6449435*n^4 + 3292426*n^3 - 41693904*n^2 + 63681552*n - 24491808)*a(n-1) + 3*n*(2451*n^8 - 73161*n^7 + 850153*n^6 - 4796076*n^5 + 12712261*n^4 - 7403931*n^3 - 33886709*n^2 + 64848252*n - 30495792)*a(n-2) - (8170*n^9 - 256125*n^8 + 3222045*n^7 - 20734872*n^6 + 70290303*n^5 - 101053185*n^4 - 62925628*n^3 + 384515340*n^2 - 387509328*n + 86320944)*a(n-3) + 3*(4085*n^9 - 134190*n^8 + 1787518*n^7 - 12351340*n^6 + 46074358*n^5 - 78991732*n^4 - 20763151*n^3 + 311152124*n^2 - 443676900*n + 188645328)*a(n-4) - (8170*n^9 - 280635*n^8 + 3929664*n^7 - 28666521*n^6 + 113672493*n^5 - 215520840*n^4 + 17606573*n^3 + 648300408*n^2 - 951192216*n + 363243312)*a(n-5) + 2*(4085*n^9 - 146445*n^8 + 2159949*n^7 - 16771674*n^6 + 71463813*n^5 - 145058547*n^4 - 9273941*n^3 + 640553178*n^2 - 1114925472*n + 598040712)*a(n-6) + (8170*n^9 - 305145*n^8 + 4669113*n^7 - 37343346*n^6 + 161916525*n^5 - 325736907*n^4 - 55373986*n^3 + 1484026824*n^2 - 2345628420*n + 1080273456)*a(n-7) + (6536*n^9 - 253920*n^8 + 4039503*n^7 - 33528057*n^6 + 150519924*n^5 - 315037869*n^4 - 26105741*n^3 + 1400728128*n^2 - 2351058696*n + 1235710944)*a(n-8) + (n-9)*(6536*n^8 - 204900*n^7 + 2511339*n^6 - 14959584*n^5 + 41778954*n^4 - 25451829*n^3 - 129319352*n^2 + 282520572*n - 168563664)*a(n-9) + 3*(n-10)*(n-9)*(817*n^7 - 18668*n^6 + 155929*n^5 - 559001*n^4 + 589888*n^3 + 1351597*n^2 - 3752130*n + 2343528)*a(n-10). - Vaclav Kotesovec, Jun 05 2014

a(n) ~ c * d^n / n^(3/2), where d = 3.8821590268628506747194368909643384060073824... is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c = 0.56162811676670317653498040062091920282038218... . - Vaclav Kotesovec, Jun 05 2014