A216318 - OEIS

%I #24 Oct 30 2020 12:20:14

%S 0,1,2,8,31,119,456,1749,6721,25883,99892,386308,1496782,5809478,

%T 22584160,87922215,342741285,1337698515,5226732060,20442936360,

%U 80031775890,313585934610,1229695855440,4825705232010,18950613058026,74467158658974,292797216620776,1151895428382104

%N Number of peaks in all Dyck n-paths after changing each valley to a peak by the transform DU -> UD.

%H Vincenzo Librandi, <a href="/A216318/b216318.txt">Table of n, a(n) for n = 0..1000</a>

%F a(0)=0, a(1)=1, a(n>=2) = A001700(n-1) - Sum_{k=0..n-3} A001700(k) + Sum_{k=0..n-2} A003516(k) - 1.

%F G.f.: (16*x*(1+sqrt(1-4*x)+(5+3*sqrt(1-4*x)-2*x) * (-1+x)*x)) / ((1+sqrt(1-4*x))^5 * sqrt(1-4*x)).

%F a(n) ~ 5*2^(2*n-3)/sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 21 2014

%F a(n) = C(2*n-2,n-1)*(5*(n-1)^2+5*(n-1)+2)/(2*n*(n+1)), n>1, a(0)=0, a(1)=1. - _Vladimir Kruchinin_, Oct 30 2020

%e The 5 Dyck 3-paths after changing DU to UD become two copies of UUUDDD with one peak each and three copies of UUDUDD with two peaks each giving a(3)=8.

%t CoefficientList[Series[(16*x*(1+Sqrt[1-4*x]+(5+3*Sqrt[1-4*x]-2*x)*(-1+x) x))/((1+Sqrt[1-4*x])^5*Sqrt[1-4*x]),{x,0,27}],x]

%o (PARI) x='x+O('x^50); concat([0], Vec((16*x*(1+sqrt(1-4*x)-(5+3*sqrt(1-4*x)-2*x)*(1-x)*x)) / ((1+sqrt(1-4*x))^5*sqrt(1-4*x)))) \\ _G. C. Greubel_, Apr 01 2017

%o (Maxima)

%o a(n):=if n<2 then n else binomial(2*n-2,n-1)*(5*(n-1)^2+5*(n-1)+2)/(2*n*(n+1)); /* _Vladimir Kruchinin_, Oct 30 2020 */

%Y Cf. A001700, A003516, A005891.

%K nonn

%O 0,3

%A _David Scambler_, Sep 03 2012