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A143955 Sum of the altitudes of the leftmost valleys of all Dyck paths of semilength n (if path has no valley, then this altitude is taken to be 0). 3

%I

%S 0,0,0,1,6,26,101,376,1377,5017,18277,66727,244377,898129,3312554,

%T 12260129,45526754,169588754,633580634,2373550184,8914719134,

%U 33562602134,126640791884,478848661898,1814142235028,6885560250148

%N Sum of the altitudes of the leftmost valleys of all Dyck paths of semilength n (if path has no valley, then this altitude is taken to be 0).

%C a(n)=Sum(k*A097607(n,k), k>=0).

%C The positive terms form the partial sums of A000344.

%H Vincenzo Librandi, <a href="/A143955/b143955.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: z^3*C^5/(1-z), where C=(1-sqrt(1-4*z))/(2*z) is the generating function of the Catalan numbers (A000108).

%F Conjecture: (n+2)*a(n) -4*(2*n+1)*a(n-1) +2*(10*n-9)*a(n-2) +17*(2-n)*a(n-3) +2*(2*n-7)*a(n-4)=0. - _R. J. Mathar_, Jul 24 2012

%F a(n) ~ 5*4^n/(3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014

%F a(n) = 5*Sum_{k=2..n-1}(binomial(2*k,k-2)/(k+3)). - _Vladimir Kruchinin_, Mar 15 2016

%e a(4)=6 because the Dyck paths of semilength 4 with leftmost valley at a positive altitude are UUDUDDUD, UUDUDUDD, UUDUUDDD, UUUDDUDD and UUUDUDDD, where U=(1,1) and D=(1,-1); these altitudes are 1, 1, 1, 1 and 2, respectively.

%p C:=((1-sqrt(1-4*z))*1/2)/z: G:=z^3*C^5/(1-z): Gser:=series(G,z=0,32): seq(coeff(Gser,z,n),n=0..27);

%t CoefficientList[Series[x^3 ((1 - (1 - 4 x)^(1/2))/(2 x))^5/(1 - x), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)

%o (Maxima)

%o a(n):=5*sum(binomial(2*k,k-2)/(k+3),k,2,n-1); /* _Vladimir Kruchinin_, Mar 15 2016 */

%Y Cf. A000108, A000344, A097607.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Oct 14 2008

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Last modified June 6 14:04 EDT 2020. Contains 334827 sequences. (Running on oeis4.)