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Number of base pyramids in all length n left factors of Dyck paths.
2

%I #18 Mar 28 2017 18:23:49

%S 0,0,1,1,4,5,14,20,49,76,175,286,637,1078,2353,4081,8788,15521,33098,

%T 59279,125476,227239,478192,873885,1830270,3370029,7030570,13027729,

%U 27088870,50469889,104647630,195892564,405187825,761615284,1571990935,2965576714,6109558585,11563073314

%N Number of base pyramids in all length n left factors of Dyck paths.

%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="/A191790/b191790.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k>=0} k*A191788(n,k).

%F G.f.: z^2*c^2/((1-z^2)*(1-z*c)), where c=(1-sqrt(1-4*z^2))/(2*z^2).

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

%F a(n) = Sum_{k=0..n/2}((Sum_{j=ceiling(n/2)-k+1..2*(n-2*k)}(((2*j+2*k-n))*binomial(n-2*k-1,n-2*k-j)/j))). - _Vladimir Kruchinin_, Mar 04 2016

%F Conjecture: (n+1)*a(n) +(-n-1)*a(n-1) +5*(-n+1)*a(n-2) +(5*n-11)*a(n-3) +2*(2*n-3)*a(n-4) +4*(-n+3)*a(n-5)=0. - _R. J. Mathar_, Jun 14 2016

%F Conjecture: -(n+1)*(n-2)*a(n) +2*(n-1)*a(n-1) +(5*n-3)*(n-2)*a(n-2) +2*(-n+1)*a(n-3) -4*(n-1)*(n-2)*a(n-4)=0. - _R. J. Mathar_, Jun 14 2016

%e a(4)=4 because in UUDU, UUUD, UUUU, (UD)(UD), (UD)UU, and (UUDD) we have 0 + 0 + 0 + 2 + 1 + 1 = 4 base pyramids (shown between parentheses).

%p c := ((1-sqrt(1-4*z^2))*1/2)/z^2: g := z^2*c^2/((1-z^2)*(1-z*c)): gser := series(g, z = 0, 40): seq(coeff(gser, z, n), n = 0 .. 37);

%t CoefficientList[Series[x^2*(((1-Sqrt[1-4*x^2])*1/2)/x^2)^2/((1-x^2)*(1-x*((1-Sqrt[1-4*x^2])*1/2)/x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 21 2014 *)

%o (Maxima)

%o a(n):=sum((sum(((2*j+2*k-n))*binomial(n-2*k-1,n-2*k-j)/j,j,ceiling(n/2)-k+1,2*(n-2*k))),k,0,n/2); /* _Vladimir Kruchinin_, Mar 04 2016 */

%o (PARI) x='x+O('x^50); concat([0,0], Vec((1-2*x^2 - sqrt(1-4*x^2))/(x*(1-x^2)*(2*x-1 + sqrt(1-4*x^2))))) \\ _G. C. Greubel_, Mar 27 2017

%Y Cf. A191788.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Jun 18 2011