OFFSET
0,5
COMMENTS
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)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{k>=0} k*A191788(n,k).
G.f.: z^2*c^2/((1-z^2)*(1-z*c)), where c=(1-sqrt(1-4*z^2))/(2*z^2).
a(n) ~ 2^(n+3/2)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 21 2014
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
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
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
EXAMPLE
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).
MAPLE
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);
MATHEMATICA
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 *)
PROG
(Maxima)
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 */
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 18 2011
STATUS
approved