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A191790 Number of base pyramids in all length n left factors of Dyck paths. 2
0, 0, 1, 1, 4, 5, 14, 20, 49, 76, 175, 286, 637, 1078, 2353, 4081, 8788, 15521, 33098, 59279, 125476, 227239, 478192, 873885, 1830270, 3370029, 7030570, 13027729, 27088870, 50469889, 104647630, 195892564, 405187825, 761615284, 1571990935, 2965576714, 6109558585, 11563073314 (list; graph; refs; listen; history; text; internal format)
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
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
Cf. A191788.
Sequence in context: A042321 A050164 A188021 * A246984 A306897 A348889
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 18 2011
STATUS
approved

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Last modified May 27 02:41 EDT 2024. Contains 372847 sequences. (Running on oeis4.)