cp's OEIS Frontend

A101499 A Chebyshev transform of the Catalan numbers.

%I A101499 #36 Nov 22 2024 06:32:14
%S A101499 1,1,1,3,9,25,73,223,697,2217,7161,23427,77457,258417,868881,2941311,
%T A101499 10016241,34289041,117935473,407344771,1412307481,4913508489,
%U A101499 17148100569,60018592735,210619695913,740910077497,2612194773481
%N A101499 A Chebyshev transform of the Catalan numbers.
%C A101499 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.
%C A101499 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
%H A101499 Vincenzo Librandi, <a href="/A101499/b101499.txt">Table of n, a(n) for n = 0..1000</a>
%H A101499 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6.
%H A101499 Jean-Luc Baril and Paul Barry, <a href="https://arxiv.org/abs/2212.12404">Two kinds of partial Motzkin paths with air pockets</a>, arXiv:2212.12404 [math.CO], 2022.
%H A101499 Jean-Luc Baril, Daniela Colmenares, José L. Ramírez, Emmanuel D. Silva, Lina M. Simbaqueba, and Diana A. Toquica, <a href="http://jl.baril.u-bourgogne.fr/bacorasisito.pdf">Consecutive pattern-avoidance in Catalan words according to the last symbol</a>, Univ. Bourgogne (France 2023).
%F A101499 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)}}}.
%F A101499 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
%F A101499 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
%F A101499 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
%F A101499 D-finite with recurrence (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
%F A101499 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
%t A101499 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 *)
%o A101499 (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 */
%Y A101499 Cf. A000108, A049310.
%K A101499 easy,nonn
%O A101499 0,4
%A A101499 _Paul Barry_, Dec 04 2004