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 A118974 Sum of the lengths of the first descents in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1). 2
 0, 0, 2, 4, 11, 31, 94, 298, 977, 3283, 11243, 39087, 137569, 489171, 1754596, 6340756, 23063731, 84372061, 310216081, 1145748061, 4248861631, 15814069951, 59054807821, 221197379221, 830819449003, 3128511421663, 11808294045071 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265. FORMULA a(n) = Sum_{k=1,..,n} k*A118972(n,k). G.f.: z^2*CF(1+C-zC)/(1-z), where F = [1-sqrt(1-4*z)]/[z*(3-sqrt(1-4*z)] and C = [1-sqrt(1-4*z)]/(2*z) is the Catalan function. a(n) ~ 17*4^n/(27*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 20 2014 Conjecture: 2*(n+1)*(17*n^2-65*n+60)*a(n) -3*(3*n-4)*(17*n^2-48*n+15)*a(n-1) +3*(17*n^3-82*n^2+121*n-60)*a(n-2) +2*(2*n-5) *(17*n^2-31*n+12) *a(n-3)=0. - R. J. Mathar, Jun 22 2016 EXAMPLE a(4)=11 because in the hill-free Dyck paths of semilength 4, namely uu(dd)uudd, uu(d)uuddd, uu(d)ududd, uuu(dd)udd, uuu(d)uddd and uuuu(dddd), the sum of the lengths of the first descents (shown between parentheses) is 2+1+1+2+1+4=11. MAPLE F:=(1-sqrt(1-4*z))/z/(3-sqrt(1-4*z)): C:=(1-sqrt(1-4*z))/2/z: g:=series(z^2*C*F*(1+C-z*C)/(1-z), z=0, 32): seq(coeff(g, z, n), n=0..28); MATHEMATICA CoefficientList[Series[x^2*(1-Sqrt[1-4*x])/2/x*(1-Sqrt[1-4*x])/x/(3-Sqrt[1-4*x])*(1+(1-Sqrt[1-4*x])/2/x-x*(1-Sqrt[1-4*x])/2/x)/(1-x), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 20 2014 *) PROG (PARI) x='x+O('x^50); concat([0, 0], Vec(x^2*(1-sqrt(1-4*x))/2/x*(1-sqrt(1-4*x))/x/(3-sqrt(1-4*x))*(1+(1-sqrt(1-4*x))/2/x-x*(1-sqrt(1-4*x))/2/x)/(1-x))) \\ G. C. Greubel, Mar 18 2017 CROSSREFS Cf. A000957, A118972. Sequence in context: A274775 A039300 A247333 * A119020 A073191 A173139 Adjacent sequences:  A118971 A118972 A118973 * A118975 A118976 A118977 KEYWORD nonn AUTHOR Emeric Deutsch, May 08 2006 STATUS approved

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Last modified August 6 14:14 EDT 2020. Contains 336246 sequences. (Running on oeis4.)