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A129164 Sum of pyramid weights in all skew Dyck paths of semilength n.

%I A129164 #40 Oct 14 2016 07:41:37
%S A129164 1,5,22,97,436,1994,9241,43257,204052,968440,4619011,22120630,
%T A129164 106300507,512321437,2475395302,11986728457,58156146652,282640193312,
%U A129164 1375737276787,6705522150972,32724071280517,159878425878847,781910419686412,3827639591654862,18753350784435811
%N A129164 Sum of pyramid weights in all skew Dyck paths of semilength n.
%C A129164 A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a skew Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a skew Dyck path (word) is the sum of the heights of its maximal pyramids.
%H A129164 Vincenzo Librandi, <a href="/A129164/b129164.txt">Table of n, a(n) for n = 1..200</a>
%H A129164 A. Denise and R. Simion, <a href="http://dx.doi.org/10.1016/0012-365X(93)E0147-V">Two combinatorial statistics on Dyck paths</a>, Discrete Math., 137, 1995, 155-176.
%H A129164 E. Deutsch, E. Munarini, S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
%F A129164 a(n) = Sum_{k=1..n} k*A129163(n,k).
%F A129164 Partial sums of A026378.
%F A129164 G.f. = [1/sqrt(1-6*z+5*z^2)-1/(1-z)]/2.
%F A129164 a(n) = n*Sum_(m=1..n, Sum_(k=m..n,(binomial(-m+2*k-1,k-1)*binomial(n-1,k-1))/k)). - _Vladimir Kruchinin_, Oct 07 2011
%F A129164 Recurrence: n*a(n) = (7*n-4)*a(n-1) - (11*n-14)*a(n-2) + 5*(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 20 2012
%F A129164 a(n) ~ 5^(n+1/2)/(4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 20 2012
%F A129164 a(n) = n*hypergeometric([1, 3/2, 1-n], [2, 2], -4). - _Peter Luschny_, Sep 16 2014
%F A129164 a(n) = Sum_{k=1..n} (binomial(n,k)*binomial(2*k,k))/2. - _Vladimir Kruchinin_, Oct 11 2016
%e A129164 a(2)=5 because the pyramid weights of the paths (UD)(UD), (UUDD) and U(UD)L are 2, 2 and 1, respectively (the maximal pyramids are shown between parentheses).
%p A129164 G:=(1/sqrt(1-6*z+5*z^2)-1/(1-z))/2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..26);
%t A129164 Rest[CoefficientList[Series[(1/Sqrt[1-6*x+5*x^2]-1/(1-x))/2, {x, 0, 20}], x]] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%t A129164 a[n_] := (Hypergeometric2F1[1/2, -n, 1, -4]-1)/2; Array[a, 25] (* _Jean-François Alcover_, Oct 11 2016, after _Vladimir Kruchinin_ *)
%o A129164 (Maxima)
%o A129164 a(n):=n*sum(sum((binomial(-m+2*k-1,k-1)*binomial(n-1,k-1))/k,k,m,n), m,1,n); /* _Vladimir Kruchinin_, Oct 07 2011 */
%o A129164 (Maxima)
%o A129164 a(n):=sum(binomial(n,k)*binomial(2*k,k),k,1,n)/2; /* _Vladimir Kruchinin_, Oct 11 2016 */
%o A129164 (Sage)
%o A129164 A129164 = lambda n: n*hypergeometric([1, 3/2, 1-n], [2, 2], -4)
%o A129164 [simplify(A129164(n)) for n in (1..25)] # _Peter Luschny_, Sep 16 2014
%Y A129164 Cf. A026378, A129163.
%K A129164 nonn
%O A129164 1,2
%A A129164 _Emeric Deutsch_, Apr 03 2007