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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).

%I A143955 #24 May 16 2022 04:37:21
%S A143955 0,0,0,1,6,26,101,376,1377,5017,18277,66727,244377,898129,3312554,
%T A143955 12260129,45526754,169588754,633580634,2373550184,8914719134,
%U A143955 33562602134,126640791884,478848661898,1814142235028,6885560250148
%N 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).
%C A143955 The positive terms form the partial sums of A000344.
%H A143955 Vincenzo Librandi, <a href="/A143955/b143955.txt">Table of n, a(n) for n = 0..200</a>
%F A143955 a(n) = Sum_{k>=0} k*A097607(n,k).
%F A143955 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 A143955 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 A143955 a(n) ~ 5*4^n/(3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 21 2014
%F A143955 a(n) = 5*Sum_{k=2..n-1}(binomial(2*k,k-2)/(k+3)). - _Vladimir Kruchinin_, Mar 15 2016
%e A143955 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 A143955 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 A143955 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 A143955 (Maxima)
%o A143955 a(n):=5*sum(binomial(2*k,k-2)/(k+3),k,2,n-1); /* _Vladimir Kruchinin_, Mar 15 2016 */
%o A143955 (Python)
%o A143955 from functools import cache
%o A143955 @cache
%o A143955 def B(n, k):
%o A143955     if n <= 0 or k <= 0: return 0
%o A143955     if n == k: return 1
%o A143955     return B(n - 1, k) + B(n, k - 1)
%o A143955 def A143955(k):
%o A143955     return B(k + 3, k - 2)
%o A143955 print([A143955(n) for n in range(26)]) # _Peter Luschny_, May 15 2022
%Y A143955 Cf. A000108, A000344, A097607, A323224.
%K A143955 nonn
%O A143955 0,5
%A A143955 _Emeric Deutsch_, Oct 14 2008