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A361306 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^4.

%I A361306 #17 Aug 24 2023 07:50:14
%S A361306 1,1,6,31,186,1191,7972,55164,391322,2830751,20801826,154853413,
%T A361306 1165316224,8850372878,67750780816,522218420336,4049564739054,
%U A361306 31570368061361,247293510244174,1945331619223591,15361731119713506,121729460653957980,967664450692965300
%N A361306 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^4.
%H A361306 Paul D. Hanna, <a href="/A361306/b361306.txt">Table of n, a(n) for n = 1..500</a>
%F A361306 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
%F A361306 (1) A(x) = Series_Reversion( x - x^2*(1+x)^4 ).
%F A361306 (2) A(x) = x + A(x)^2 * (1 + A(x))^4.
%F A361306 (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * (1+x)^(4*n) / n!.
%F A361306 (4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(4*n) / n! ).
%F A361306 (5) A(x) = x + Series_Reversion( Series_Reversion( x*(1+x)^2 ) - x^2 )^2.
%F A361306 (6) A(x) = x + Series_Reversion( -x^2 + Sum_{n>=1} (-1)^(n-1) * binomial(3*n-2,n-1) * x^n/n )^2.
%F A361306 From _Vaclav Kotesovec_, Mar 09 2023: (Start)
%F A361306 Recurrence: 3381*(n-4)*(n-3)*(n-2)*(n-1)*n*(4485934293448*n^5 - 88905588075732*n^4 + 698950092208066*n^3 - 2724285958475163*n^2 + 5263801532363671*n - 4032831805999290)*a(n) = 2*(n-4)*(n-3)*(n-2)*(n-1)*(33204885640102096*n^6 - 707886491396721408*n^5 + 6160367858867908768*n^4 - 27918165429184721124*n^3 + 69150795811214975011*n^2 - 88077097294043237943*n + 44480953779348451050)*a(n-1) + 8*(n-4)*(n-3)*(n-2)*(52772531028122272*n^7 - 1256975462235400336*n^6 + 12611049851568548176*n^5 - 69004162305753446968*n^4 + 222104765912229832762*n^3 - 419924105934755620321*n^2 + 431120275047208552290*n - 185089750933520270250)*a(n-2) + 48*(n-4)*(n-3)*(17647665510424432*n^8 - 482112074818112928*n^7 + 5693971809001104840*n^6 - 37956706633792772384*n^5 + 156126872715173363823*n^4 - 405548028261835673882*n^3 + 649232078072133939050*n^2 - 585187986606994739801*n + 227161430445970883100)*a(n-3) + 32*(n-4)*(21945190563547616*n^9 - 698268423629052336*n^8 + 9788485232517982416*n^7 - 79313303231764021176*n^6 + 409187506797434806734*n^5 - 1393249646753024170299*n^4 + 3129189249705937191544*n^3 - 4467594298222926610959*n^2 + 3676695031470911619960*n - 1327813620065788842000)*a(n-4) + 72*(2*n - 7)*(3*n - 14)*(3*n - 13)*(6*n - 31)*(6*n - 29)*(4485934293448*n^5 - 66475916608492*n^4 + 388187082839618*n^3 - 1116009867370877*n^2 + 1578887211201855*n - 878785793685000)*a(n-5).
%F A361306 a(n) ~ 1/(2 * (1 + s) * sqrt(Pi*(1 + 10*s + 15*s^2)) * n^(3/2) * r^(n - 1/2)), where r = 0.1176087332021218420455915375218722861407778043565... and s = 0.1894485384658193296593809633217117092941452563863... are real roots of the system of equations r + s^2 * (1+s)^4 = s, 2*s*(1+s)^3 * (1+3*s) = 1. (End)
%F A361306 a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(4*k,n-k)/(n+k+1). - _Seiichi Manyama_, Aug 24 2023
%e A361306 G.f.: A(x) = x + x^2 + 6*x^3 + 31*x^4 + 186*x^5 + 1191*x^6 + 7972*x^7 + 55164*x^8 + 391322*x^9 + ...
%e A361306 such that sqrt(A(x) - x) = A(x)*(1 + A(x))^2.
%e A361306 A(x)*(1 + A(x))^2 = x + 3*x^2 + 11*x^3 + 60*x^4 + 355*x^5 + 2261*x^6 + 15094*x^7 + 104208*x^8 + ...
%e A361306 A(x)*(1 + A(x))^2 = Series_Reversion( -x^2 + Sum_{n>=1} (-1)^(n-1) * binomial(3*n-2,n-1)*x^n/n ).
%o A361306 (PARI) {a(n)=polcoeff(serreverse(x - x^2*(1+x)^4 +x*O(x^n)), n)}
%o A361306 for(n=1, 30, print1(a(n), ", "))
%o A361306 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A361306 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(4*m)/m!)); polcoeff(A, n)}
%o A361306 for(n=1, 30, print1(a(n), ", "))
%o A361306 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A361306 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*(1+x+x*O(x^n))^(4*m)/m!))); polcoeff(A, n)}
%o A361306 for(n=1, 30, print1(a(n), ", "))
%Y A361306 Cf. A361304, A361305, A214372, A006013.
%K A361306 nonn
%O A361306 1,3
%A A361306 _Paul D. Hanna_, Mar 08 2023