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%I #45 Aug 05 2023 02:11:48
%S 1,2,7,28,123,572,2769,13806,70414,365636,1926505,10273870,55349155,
%T 300783420,1646828655,9075674700,50304255210,280248358964,
%U 1568399676946,8813424968192,49709017472751,281306750922072,1596802663432503
%N G.f. satisfies: A(x) = (1 + x*A(x) + x^2*A(x)^2)^2.
%H G. C. Greubel, <a href="/A143927/b143927.txt">Table of n, a(n) for n = 0..1000</a>
%H Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F Self-convolution of A006605.
%F Bisection of A143926.
%F a(n) = ((24*n+12)*A006605(n) + (3*n+5)*A006605(n+1))/(13*n+17). - _Mark van Hoeij_, Jul 14 2010
%F a(n) = (1/(n+1))*Sum_{j=0..2*n+2} (binomial(j,2*j-3*n-4)*binomial(2*n+2 ,j)). - _Vladimir Kruchinin_, Dec 24 2010
%F a(n) = GegenbauerPoly(n,-2n-2,-1/2)/(n+1). - _Emanuele Munarini_, Oct 20 2016
%F a(n) = T(2*n+2, n)/(n+1), where T(n,k) = A027907(n,k). - _Emanuele Munarini_, Oct 20 2016
%F The g.f. A(x) satisfies 1 + x*A'(x)/A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 266*x^3 + ..., the g.f. of A027908. - _Peter Bala_, Aug 03 2023
%t Table[GegenbauerC[n,-2n-2,-1/2]/(n+1),{n,0,12}] (* _Emanuele Munarini_, Oct 20 2016 *)
%t n = 20;
%t A = Sum[a[k] x^k, {k, 0, n}] + x O[x]^n;
%t Table[a[k], {k, 0, n}] /. Reverse[Solve[LogicalExpand[(1 + x A + x^2 A^2)^2 == A]]] (* _Emanuele Munarini_, Oct 20 2016 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=(1+x*A+x^2*A^2)^2);polcoeff(A,n)}
%o (Maxima) makelist(ultraspherical(n,-2*n-2,-1/2)/(n+1),n,0,12); /* _Emanuele Munarini_, Oct 20 2016 */
%Y Cf. A006605, A027908, A143926.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 08 2008
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