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%I
%S 1,1,3,11,44,185,804,3579,16229,74690,347984,1638169,7780876,37245028,
%T 179503340,870374211,4243141332,20786340271,102275718924,505235129250,
%U 2504876652190,12459922302900,62167152967680,311040862133625
%N Expansion of c(x*c(x)), where c(x) is the g.f. of A000108.
%C Old name was: Expansion of 1/(1 - x*c(x) * c(x*c(x))), where c(x) is the g.f. of A000108.
%C Row sums of number triangle A127631.
%C Hankel transform appears to be A075845.
%C Catalan transform of Catalan numbers . - _Philippe DELEHAM_, Jun 20 2007
%C Number of functions f:[1,n] -> [1,n] satisfying the condition that, for all i<j, f(j)-(j-i) is not in the interval [1,f(i)-1]; see the Callan reference. [_Joerg Arndt_, May 31 2013]
%H David Callan, <a href="http://arxiv.org/abs/1111.0996">A combinatorial interpretation of the Catalan transform of the Catalan numbers</a>, arXiv:1111.0996, 2011.
%H Y. Sun, Z. Wang, <a href="http://dx.doi.org/10.1007/s00373-010-0950-9">Consecutive pattern avoidances in non-crossing trees</a>, Graph. Combinat. 26 (2010) 815-832, table 1, {ud}
%F a(n) = A127714(n+1,2n+1).
%F G.f. A(x) satisfies 0 = 1 - A(x) + A(x)^2 * x * c(x) where c(x) is the g.f. of A000108.
%F G.f.: 2/(1 + sqrt( 2 * sqrt(1 -4*x) - 1)). - _Michael Somos_, May 04 2007
%F a(n) = Sum_{k, 0<=k<=n}A106566(n,k)*A000108(k). - _Philippe DELEHAM_, Jun 20 2007
%F a(n) = sum(m=1..n, m*sum(k=m..n, binomial(2*k-m-1,k-1)*binomial(2*n-k-1,n-1)))/n, a(0)=1. [_Vladimir Kruchinin_, Oct 08 2011]
%t a[n_] := Sum[m*(2*n-m-1)!*HypergeometricPFQ[{m/2+1/2, m/2, m-n}, {m, m-2*n+1}, 4]/(n!*(n-m)!), {m, 1, n}]; a[0]=1; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Jul 24 2012, after _Vladimir Kruchinin_ *)
%o (PARI) {a(n)= if(n<1, n==0, polcoeff( serreverse( x*(1-x)^3*(1-x^3)/(1-x^2)^4 +x*O(x^n) ), n))} /* _Michael Somos_, May 04 2007 */
%o (PARI) {a(n)= local(A); if(n<1, n==0, A= serreverse( x-x^2 +x*O(x^n) ); polcoeff( 1/(1 - subst(A, x, A)), n))} /* _Michael Somos_, May 04 2007 */
%o (Maxima) a(n):=if n=0 then 1 else sum(m*sum(binomial(2*k-m-1,k-1)*binomial(2*n-k-1,n-1),k,m,n),m,1,n)/n; [_Vladimir Kruchinin_, Oct 08 2011]
%Y Cf. A127714.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Jan 20 2007, Jan 25 2007
%E Better name by _David Callan_, Jun 03 2013
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