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A361305 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^3.

%I A361305 #14 Aug 24 2023 07:50:10
%S A361305 1,1,5,23,123,700,4170,25677,162101,1043603,6825429,45222437,
%T A361305 302892681,2047499720,13950769772,95710823820,660609751890,
%U A361305 4584018016679,31960334260971,223782306725768,1572921720684820,11094267854522250,78499108540111380,557041048588402170
%N A361305 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^3.
%H A361305 Paul D. Hanna, <a href="/A361305/b361305.txt">Table of n, a(n) for n = 1..500</a>
%F A361305 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F A361305 (1) A(x) = Series_Reversion( x - x^2*(1+x)^3 ).
%F A361305 (2) A(x) = x + A(x)^2*(1 + A(x))^3.
%F A361305 (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * (1+x)^(3*n) / n!.
%F A361305 (4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(3*n) / n! ).
%F A361305 (5) A(x) = x + Series_Reversion( Series_Reversion( x*(1+x)^(3/2) ) - x^2 )^2.
%F A361305 From _Vaclav Kotesovec_, Mar 09 2023: (Start)
%F A361305 Recurrence: 283*(n-3)*(n-2)*(n-1)*n*(3869140*n^3 - 39431172*n^2 + 133221959*n - 149076999)*a(n) = 4*(n-3)*(n-2)*(n-1)*(1199433400*n^4 - 14022813420*n^3 + 59620648652*n^2 - 107988096753*n + 68872774500)*a(n-1) + 6*(n-3)*(n-2)*(3718243540*n^5 - 52766330452*n^4 + 294066223701*n^3 - 803084308634*n^2 + 1072900001465*n - 559958090580)*a(n-2) + 12*(n-3)*(2453034760*n^6 - 43397123748*n^5 + 316599139024*n^4 - 1218191215329*n^3 + 2605017314614*n^2 - 2932345787601*n + 1355713586640)*a(n-3) + 5*(5*n - 21)*(5*n - 19)*(5*n - 18)*(5*n - 17)*(3869140*n^3 - 27823752*n^2 + 65967035*n - 51417072)*a(n-4).
%F A361305 a(n) ~ 1/(2 * sqrt(Pi*(1 + s)*(1 + 8*s + 10*s^2)) * n^(3/2) * r^(n - 1/2)), where r = 0.1321273811013026086255933373480102325835852282463... and s = 0.2180852364825231879900920777342190033594997222087... are real roots of the system of equations r + s^2 * (1+s)^3 = s, s * (1+s)^2 * (2+5*s) = 1. (End)
%F A361305 a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(3*k,n-k)/(n+k+1). - _Seiichi Manyama_, Aug 24 2023
%e A361305 G.f.: A(x) = x + x^2 + 5*x^3 + 23*x^4 + 123*x^5 + 700*x^6 + 4170*x^7 + 25677*x^8 + 162101*x^9 + 1043603*x^10 + ...
%e A361305 such that A(x) = x + A(x)^2 * (1 + A(x))^3.
%e A361305 Related series.
%e A361305 A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 56*x^5 + 317*x^6 + 1876*x^7 + 11499*x^8 + 72352*x^9 + 464585*x^10 + ...
%e A361305 (1 + A(x))^3 = 1 + 3*x + 6*x^2 + 22*x^3 + 105*x^4 + 555*x^5 + 3151*x^6 + 18735*x^7 + 115200*x^8 + 726530*x^9 + ...
%o A361305 (PARI) {a(n)=polcoeff(serreverse(x-x^2*(1+x)^3+x*O(x^n)), n)}
%o A361305 for(n=1, 30, print1(a(n), ", "))
%o A361305 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A361305 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(3*m)/m!)); polcoeff(A, n)}
%o A361305 for(n=1, 30, print1(a(n), ", "))
%o A361305 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A361305 {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))^(3*m)/m!))); polcoeff(A, n)}
%o A361305 for(n=1, 30, print1(a(n), ", "))
%Y A361305 Cf. A361304, A361306, A214372.
%K A361305 nonn
%O A361305 1,3
%A A361305 _Paul D. Hanna_, Mar 08 2023