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A274734 G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x*A(x)^2).

%I A274734 #21 Jan 27 2024 07:41:19
%S A274734 1,3,15,94,661,4983,39363,321587,2694860,23035341,200068651,
%T A274734 1760558682,15663027711,140648129383,1273083938979,11603500739475,
%U A274734 106404140837773,980977232554344,9087285865886766,84541177049414342,789545725457924023,7399515198155161271,69568021610270590583,655960254857760518109,6201585037793334756198,58775103307105512895151
%N A274734 G.f. satisfies A(x) = (1 + x*A(x))^2 * (1 + x*A(x)^2).
%C A274734 More generally, if G(x) satisfies
%C A274734 G(x) = (1 + a*x*G(x))^m * (1 + b*x*G(x)^2), then
%C A274734 G(x) = (1/x) * Series_Reversion( x * (1 - b*x*(1 + a*x)^m) / (1 + a*x)^m ).
%H A274734 Vaclav Kotesovec, <a href="/A274734/b274734.txt">Table of n, a(n) for n = 0..400</a>
%F A274734 G.f.: (1/x) * Series_Reversion( x * (1 - x*(1+x)^2) / (1+x)^2 ).
%F A274734 Recurrence: 31*(n-1)*n*(n+1)*(5974*n^3 - 40359*n^2 + 90115*n - 67124)*a(n) = 2*(n-1)*n*(1003632*n^4 - 7282128*n^3 + 18518502*n^2 - 18822839*n + 5649607)*a(n-1) - 2*(n-1)*(740776*n^5 - 6486068*n^4 + 21715762*n^3 - 34616651*n^2 + 26123385*n - 7413210)*a(n-2) + 2*(2*n - 5)*(65714*n^5 - 575377*n^4 + 1957337*n^3 - 3264653*n^2 + 2726129*n - 941430)*a(n-3) + 4*(n-3)*(2*n - 7)*(2*n - 5)*(5974*n^3 - 22437*n^2 + 27319*n - 11394)*a(n-4). - _Vaclav Kotesovec_, Nov 18 2017
%F A274734 a(n) ~ sqrt((s*(1+r*s)*(2 + s + 3*r*s^2)) / (1 + r*(1 + 6*s*(1+r*s)))) / (2*sqrt(Pi) * n^(3/2) * r^n), where r = 0.099424837262345547872398211374352678... and s = 2.183663565361369673488934371066403742... are roots of the system of equations (1 + r*s)^2*(1 + r*s^2) = s, 2*r*(1 + s + 2*r^2*s^3 + r*s*(1 + 3*s)) = 1. - _Vaclav Kotesovec_, Nov 18 2017
%F A274734 a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(n+k,k) * binomial(2*n+2*k+2,n-k). - _Seiichi Manyama_, Jan 27 2024
%e A274734 G.f.: A(x) = 1 + 3*x + 15*x^2 + 94*x^3 + 661*x^4 + 4983*x^5 + 39363*x^6 + 321587*x^7 +...
%o A274734 (PARI) {a(n) = my(A=1); for(i=1,n, A = (1 + x*A)^2 * (1 + x*A^2) + x*O(x^n) ); polcoeff(A,n)}
%o A274734 for(n=0,30,print1(a(n),", "))
%o A274734 (PARI) {a(n) = my(A=1); A = (1/x)*serreverse(x*(1-x*(1+x)^2)/(1+x +x^2*O(x^n) )^2 ); polcoeff(A,n)}
%o A274734 for(n=0,30,print1(a(n),", "))
%Y A274734 Cf. A181734, A274735.
%K A274734 nonn
%O A274734 0,2
%A A274734 _Paul D. Hanna_, Aug 02 2016