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 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 Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6. 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 Cf. A000108, A049310. Sequence in context: A211282 A211298 A138574 * A004665 A196431 A244826 Adjacent sequences: A101496 A101497 A101498 * A101500 A101501 A101502 KEYWORD easy,nonn AUTHOR Paul Barry, Dec 04 2004 STATUS approved

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