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%I #39 Jul 24 2022 13:10:38
%S 0,0,1,2,6,12,28,56,121,242,507,1014,2093,4186,8569,17138,34902,69804,
%T 141664,283328,573574,1147148,2318010,4636020,9354540,18709080,
%U 37708672,75417344,151868100,303736200,611180252,1222360504,2458123705,4916247410,9881255187
%N Number of base pyramids in all dispersed Dyck paths of length n (i.e., in all Motzkin paths of length n with no (1,0)-steps at positive heights).
%C A base pyramid is a factor of the form U^j D^j (j>0), starting at the horizontal axis. Here U=(1,1) and D=(1,-1).
%H G. C. Greubel, <a href="/A191394/b191394.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = Sum_{k=0..(1 + ceiling(n/2))} k*A191392(n, k), formula clarified by _G. C. Greubel_.
%F G.f.: 4*x^2/((1-x^2)*(1-2*x+sqrt(1-4*x^2))^2).
%F a(n) ~ 2^(n+1)/3 * (1-sqrt(2)/sqrt(Pi*n)). - _Vaclav Kotesovec_, Mar 21 2014
%F a(n) = ceiling(2*(2^n-1)/3) - Sum_{i=1..(n+1)/2} binomial(n-2*i+1, floor((n-2*i+1)/2)) = A000975(n) - A174783(n). - _Vladimir Kruchinin_, Mar 15 2016
%F D-finite with recurrence n*a(n) -2*n*a(n-1) +(-5*n+12)*a(n-2) +2*(5*n-12)*a(n-3) +4*(n-3)*a(n-4) +8*(-n+3)*a(n-5)=0. - _R. J. Mathar_, Jun 14 2016
%e a(4) = 6 because in HHHH, HH(UD), H(UD)H, (UD)HH, (UD)(UD), and (UUDD) we have a total of 0+1+1+1+2+1 = 6 base pyramids (shown between parentheses).
%p G := 4*z^2/((1-z^2)*(1-2*z+sqrt(1-4*z^2))^2): Gser := series(G, z = 0, 38): seq(coeff(Gser, z, n), n = 0 .. 34);
%t CoefficientList[Series[(4x^2)/((1-x^2)(1-2x+Sqrt[1-4x^2])^2), {x,0,40}], x] (* _Harvey P. Dale_, Jun 19 2011 *)
%o (Maxima) a(n):=ceiling(2*(2^n-1)/3)-sum((binomial(n-2*i+1,floor((n-2*i+1)/2))),i,1,(n+1)/2); /* _Vladimir Kruchinin_, Mar 15 2016 */
%o (PARI) x='x+O('x^50); concat([0,0], Vec(4*x^2/((1-x^2)*(1-2*x+sqrt(1-4*x^2))^2))) \\ _G. C. Greubel_, Mar 26 2017
%Y Cf. A000975, A174783, A191392.
%K nonn
%O 0,4
%A _Emeric Deutsch_, Jun 04 2011
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