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%I #38 Aug 30 2021 10:44:39
%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
%F Conjecture D-finite with recurrence 1053*n*(n-1)*(n-2)*(n-3)*a(n) -36*(n-1)*(n-2)*(n-3)*(634*n-1367)*a(n-1) +24*(n-2)*(n-3)*(7966*n^2-43500*n+61181)*a(n-2) -8*(n-3)*(96128*n^3-957424*n^2+3221878*n-3665189)*a(n-3) +16*(91904*n^4-1446528*n^3+8575792*n^2-22703688*n+22652013)*a(n-4) -256*(8*n-35)*(8*n-41)*(8*n-39)*(8*n-37)*a(n-5)=0. - _R. J. Mathar_, Aug 30 2021
%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 (2nd), A158825, A158827 (4th), A158828, A158829.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Mar 28 2009
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