OFFSET
1,2
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..160
Emeric Deutsch, Problem 10658: Another Type of Lattice Path, American Math. Monthly, 107, 2000, 368-370.
FORMULA
G.f.=zA^2/(1-z), where A=1+zA^2+zA^3=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
G.f. y(x) satisfies: x*(1+y-x*y)^2 = (1-x)*y*(1-y+x*y)^2. - Vaclav Kotesovec, Mar 17 2014
a(n) ~ sqrt(23*sqrt(5)-15) * (11+5*sqrt(5))^n / (11* sqrt(5*Pi) * n^(3/2) * 2^(n+1/2)). - Vaclav Kotesovec, Mar 17 2014
D-finite with recurrence n*(2*n-1)*a(n) -6*(n-1)*(5*n-6)*a(n-1) +4*(23*n^2-97*n+111)*a(n-2) +2*(-29*n^2+142*n-174)*a(n-3) -3*(2*n-5)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022
EXAMPLE
a(2)=5 because in the A027307(2)=10 paths we have altogether 5 pyramids of the first kind (shown between parentheses): (ud)(ud), (ud)Udd, (uudd), uUddd, Udd(ud), UddUdd, Ududd, UdUddd, Uuddd, UUdddd.
MAPLE
A:=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3: g:=z*A^2/(1-z): gser:=series(g, z=0, 25): seq(coeff(gser, z^n), n=1..22);
PROG
(PARI) {a(n)=local(y=x); for(i=1, n, y=x*(1+y-x*y)^2/((1-x)*(1-y+x*y)^2) + (O(x^n))^3); polcoeff(y, n)}
for(n=1, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 11 2005
STATUS
approved