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%I #17 Apr 06 2019 09:29:32
%S 0,0,2,4,11,31,94,298,977,3283,11243,39087,137569,489171,1754596,
%T 6340756,23063731,84372061,310216081,1145748061,4248861631,
%U 15814069951,59054807821,221197379221,830819449003,3128511421663,11808294045071
%N 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).
%H G. C. Greubel, <a href="/A118974/b118974.txt">Table of n, a(n) for n = 0..1000</a>
%H E. Deutsch and L. Shapiro, <a href="https://doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%F a(n) = Sum_{k=1,..,n} k*A118972(n,k).
%F 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.
%F a(n) ~ 17*4^n/(27*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F 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
%e 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.
%p 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);
%t 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 *)
%o (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
%Y Cf. A000957, A118972.
%K nonn
%O 0,3
%A _Emeric Deutsch_, May 08 2006
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