The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A101499 A Chebyshev transform of the Catalan numbers. 4
1, 1, 1, 3, 9, 25, 73, 223, 697, 2217, 7161, 23427, 77457, 258417, 868881, 2941311, 10016241, 34289041, 117935473, 407344771, 1412307481, 4913508489, 17148100569, 60018592735, 210619695913, 740910077497, 2612194773481 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A Chebyshev transform of A000108. Under the Chebyshev transform, we map a g.f. g(x) to (1/(1+x^2))g(x/(1+x^2)). Also equivalent to a Catalan transform followed by the Chebyshev transform to 1/(1-x), where the Catalan transform maps h(x)->h(xc(x)), c(x) the g.f. of A000108.
a(n) is the number of peakless Motzkin paths of length n in which the (1,0)-steps at level >=1 come in 2 colors. Example: a(4)=9 because, denoting u=(1,1), h=(1,0), and d=(1,-1), we have 1 path of shape hhhh, 2 paths of shape huhd, 2 paths of shape uhdh, and 2^2=4 paths of shape uhhd. - Emeric Deutsch, May 03 2011
LINKS
Jean-Luc Baril and Paul Barry, Two kinds of partial Motzkin paths with air pockets, arXiv:2212.12404 [math.CO], 2022.
Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, Consecutive pattern-avoidance in Catalan words according to the last symbol, Univ. Bourgogne (France 2023).
FORMULA
G.f.: (sqrt(1+x^2)-sqrt(1-4x+x^2))/(2x*sqrt(1+x^2)); a(n)=sum{k=0..floor(n/2), binomial(n-k, k)C(n-2k)}; a(n)=sum{k=0..floor(n/2), sum{i=0..n-2k, sum{j=0..n-2k, ((2i+1)/(n-2k+i+1))(-1)^(i-j)C(2n-4k, n-2k-i)C(i, j)}}}.
Given g.f. A(x) then B(x)=x*A(x) satisfies 0=f(x, B(x)) where f(x, y)= x-(1+x^2)*(y-y^2) . - Michael Somos, Sep 18 2006
Given g.f. A(x) then B(x)=x*A(x) satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)= w -v^2*w^2 -(1-v)*w*(v+w) +(u-u^2)^2*(v^2+w^2-v-w). - Michael Somos, Sep 18 2006
Given g.f. A(x) then B(x)=x*A(x) satisfies 0=f(B(x), B(x^2)) where f(u, v)= (v-v^2) -(u-u^2)^2*(1+2*(v-v^2)). - Michael Somos, Sep 18 2006
Conjecture: +(n+1)*a(n) +2*(-2*n+1)*a(n-1) +2*(n-1)*a(n-2) +2*(-2*n+3)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Nov 16 2012
a(n) ~ (5+3*sqrt(3)) * sqrt(2*sqrt(3)-3) * (2 + sqrt(3))^n / (8 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 08 2014
MATHEMATICA
CoefficientList[Series[(Sqrt[1+x^2]-Sqrt[1-4*x+x^2])/(2*x*Sqrt[1+x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)
PROG
(PARI) {a(n)=local(A); if(n<0, 0, n++; A=serreverse(x-x^2+x*O(x^n)); polcoeff( subst(A, x, x/(1+x^2)), n))} /* Michael Somos, Sep 18 2006 */
CROSSREFS
Sequence in context: A211282 A211298 A138574 * A004665 A196431 A244826
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 04 2004
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 26 18:00 EDT 2024. Contains 372840 sequences. (Running on oeis4.)