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A158826 Third iteration of x*C(x) where C(x) is the Catalan function (A000108). 6

%I

%S 1,3,12,54,260,1310,6824,36478,199094,1105478,6227712,35520498,

%T 204773400,1191572004,6990859416,41313818217,245735825082,

%U 1470125583756,8840948601024,53417237877396,324123222435804,1974317194619712

%N Third iteration of x*C(x) where C(x) is the Catalan function (A000108).

%C Series reversion of x - 3*x^2 + 6*x^3 - 9*x^4 + 10*x^5 - 8*x^6 + 4*x^7 - x^8. - _Benedict W. J. Irwin_, Oct 19 2016

%C Column 1 of A106566^3 (see Barry, Section 3). - _Peter Bala_, Apr 11 2017

%H G. C. Greubel, <a href="/A158826/b158826.txt">Table of n, a(n) for n = 1..1000</a>

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5, pp. 1-24.

%H Elżbieta Liszewska, Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.

%F a(n) = (1/n)*Sum_{k=1..n} [ binomial(2*k-2,k-1)*Sum_{i=k..n}( binomial(-k+2*i-1,i-1)*binomial(2*n-i-1,n-1) ) ]. - _Vladimir Kruchinin_, Jan 24 2013

%F G.f.: (1 - sqrt(-1 + 2*sqrt(-1 + 2*sqrt(1 - 4*x))))/2. - _Benedict W. J. Irwin_, Oct 19 2016

%F a(n) ~ 2^(8*n - 3) / (sqrt(5*Pi) * n^(3/2) * 39^(n - 1/2)). - _Vaclav Kotesovec_, Jul 20 2019

%t max = 22; c[x_] := Sum[ CatalanNumber[n]*x^n, {n, 0, max}]; f[x_] := x*c[x]; CoefficientList[ Series[ f@f@f@x, {x, 0, max}], x] // Rest (* _Jean-François Alcover_, Jan 24 2013 *)

%t Rest@CoefficientList[InverseSeries[x-3x^2+6x^3-9x^4+10x^5-8x^6+4x^7-x^8+O[x]^30], x] (* _Benedict W. J. Irwin_, Oct 19 2016 *)

%o (PARI) a(n)=local(F=serreverse(x-x^2+O(x^(n+1))),G=x); for(i=1,3,G=subst(F,x,G)); polcoeff(G,n)

%o (Maxima)

%o a(n):=sum(binomial(2*k-2,k-1)*sum(binomial(-k+2*i-1,i-1)*binomial(2*n-i-1,n-1),i,k,n),k,1,n)/n; // _Vladimir Kruchinin_, Jan 24 2013

%o (Python)

%o from sympy import binomial as C

%o def a(n):

%o return sum(C(2*k - 2, k - 1) * sum(C(-k + 2*i - 1, i - 1) * C(2*n - i - 1, n - 1) for i in range(k, n + 1)) for k in range(1, n + 1)) / n

%o [a(n) for n in range(1, 51)] # _Indranil Ghosh_, Apr 12 2017

%Y Cf. A121988, A158825, A158827, A158828, A158829.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Mar 28 2009

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Last modified June 14 15:19 EDT 2021. Contains 345025 sequences. (Running on oeis4.)