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%I #16 Jul 24 2022 14:35:20
%S 0,0,2,3,8,19,44,106,257,628,1549,3844,9588,24020,60391,152298,385085,
%T 975904,2478129,6303861,16060946,40977605,104682165,267730426,
%U 685451776,1756593392,4505537267,11565724164,29711413595,76379060176,196473781247
%N Sum of pyramid weights of all Dyck paths of semilength n that have no ascents and no descents of length 1.
%C A pyramid in a Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a 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 Dyck path (word) is the sum of the heights of its maximal pyramids.
%H G. C. Greubel, <a href="/A166302/b166302.txt">Table of n, a(n) for n = 0..1000</a>
%H 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.
%F a(n) = Sum_{k=0..n} k*A166301(n,k).
%F G.f.: z*(2-z)*[1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2))]/[2*(1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))].
%F a(n) ~ (3+sqrt(5))^(n+1/2) / (5^(1/4) * sqrt(Pi*n) * 2^(n+3/2)). - _Vaclav Kotesovec_, Mar 20 2014
%F D-finite with recurrence 2*(n-1)*(2494*n-8185)*a(n) +23*(-770*n^2+3867*n-3959)*a(n-1) +(13226*n^2-83741*n+101091)*a(n-2) +(-7734*n^2+51213*n-51521)*a(n-3) +(17710*n^2-114385*n+144471)*a(n-4) +(-13226*n^2+104701*n-162397)*a(n-5) +(2746*n-6149)*(n-7)*a(n-6)=0. - _R. J. Mathar_, Jul 24 2022
%e a(5)=19 because the pyramid weights of (UUDD)(UUUDDD), (UUUDDD)(UUDD), U(UUDD)(UUDD)D, and (UUUUUDDDDD) are 5, 5, 4, and 5, respectively (the maximal pyramids are shown between parentheses).
%p G := (1/2)*z*(2-z)*(1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/((1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))): Gser := series(G, z = 0, 35): seq(coeff(Gser, z, n), n = 0 .. 32);
%t CoefficientList[Series[1/2*x*(2-x)*(1+x-x^2-Sqrt[(1+x+x^2)*(1-3*x+x^2)]) /((1-x)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 20 2014 *)
%o (PARI) z='z+O('z^50); concat([0,0], Vec(z*(2-z)*(1+z-z^2-sqrt((1+z+z^2)*(1-3*z+z^2)))/(2*(1-z)*sqrt((1+z+z^2)*(1-3*z+z^2))))) \\ _G. C. Greubel_, Mar 22 2017
%Y Cf. A166301, A091866.
%K nonn
%O 0,3
%A _Emeric Deutsch_, Nov 07 2009
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