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A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(4*x*A(x)).

%I A366234 #21 Mar 28 2025 03:26:45
%S A366234 1,1,10,126,2392,60600,1916304,72917488,3246171520,165609099648,
%T A366234 9529240349440,610657739172096,43136025287678976,3330356645773880320,
%U A366234 279024535906794539008,25214258236430338160640,2444656672390982922502144,253144081975231633923342336
%N A366234 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(4*x*A(x)).
%C A366234 Related identities which hold formally for all Maclaurin series F(x):
%C A366234 (1) F(x) = (1/x) * Sum_{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),
%C A366234 (2) F(x) = (2/x) * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+1)*x*F(x)),
%C A366234 (3) F(x) = (3/x) * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * F(x)^n * exp(-(n+2)*x*F(x)),
%C A366234 (4) F(x) = (4/x) * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * F(x)^n * exp(-(n+3)*x*F(x)),
%C A366234 (5) F(x) = (k/x) * Sum_{n>=1} n*(n+k-1)^(n-2) * x^n/n! * F(x)^n * exp(-(n+k-1)*x*F(x)) for all fixed nonzero k.
%F A366234 a(n) = n! * Sum_{k=0..n} binomial(n+1, n-k)/(n+1) * 4^k * (n-k)^k / k!.
%F A366234 Let A(x)^m = Sum_{n>=0} a(n,m) * x^n/n! then a(n,m) = n!*Sum_{k=0..n} binomial(n+m, n-k)*m/(n+m) * 4^k * (n-k)^k/k!.
%F A366234 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
%F A366234 (1) A(x) = 1 + x*A(x) * exp(4*x*A(x)).
%F A366234 (2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(4*x)) ).
%F A366234 (3) A( x/(1 + x*exp(4*x)) ) = 1 + x*exp(4*x).
%F A366234 (4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-4)*x*A(x)) for all fixed nonnegative m.
%F A366234 (4.a) A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-4)*x*A(x)).
%F A366234 (4.b) A(x) = 1 + 2 * Sum_{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).
%F A366234 (4.c) A(x) = 1 + 3 * Sum_{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).
%F A366234 (4.d) A(x) = 1 + 4 * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).
%F A366234 (4.e) A(x) = 1 + 5 * Sum_{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).
%F A366234 (4.f) A(x) = 1 + 6 * Sum_{n>=1} n*(n+5)^(n-2) * x^n/n! * A(x)^n * exp(-(n+1)*x*A(x)).
%F A366234 a(n) ~ (1 + 2*LambertW(1))^(n + 3/2) * n^(n-1) / (4 * sqrt(1 + LambertW(1)) * exp(n) * LambertW(1)^(2*n + 3/2)). - _Vaclav Kotesovec_, Oct 06 2023
%e A366234 E.g.f.: A(x) = 1 + x + 10*x^2/2! + 126*x^3/3! + 2392*x^4/4! + 60600*x^5/5! + 1916304*x^6/6! + 72917488*x^7/7! + 3246171520*x^8/8! + ...
%e A366234 where A(x) satisfies A(x) = 1 + x*A(x) * exp(4*x*A(x))
%e A366234 also
%e A366234 A(x) = 1 + 1^0*x*A(x)*exp(+3*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+2*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(+1*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-0*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-1*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-2*x*A(x))/6! + ...
%e A366234 and
%e A366234 A(x) = 1 + 5*1*5^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 5*2*6^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 5*3*7^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 5*4*8^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 5*5*9^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
%e A366234 RELATED SERIES.
%e A366234 exp(x*A(x)) = 1 + x + 3*x^2/2! + 37*x^3/3! + 649*x^4/4! + 15461*x^5/5! + 471571*x^6/6! + ... + A245265(n)*x^n/n! + ...
%t A366234 nmax = 20; A[_] = 0; Do[A[x_] = 1 + x*A[x] * E^(4*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* _Vaclav Kotesovec_, Oct 06 2023 *)
%o A366234 (PARI)  /* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
%o A366234 {a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 4^k * (n-k)^k/k!)}
%o A366234 for(n=0,20,print1(a(n),", "))
%o A366234 (PARI) {a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(4*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
%o A366234 for(n=0,20,print1(a(n),", "))
%Y A366234 Cf. A161633, A366232, A366233, A366235.
%Y A366234 Cf. A365774 (dual), A245265 (exp(x*A(x))).
%K A366234 nonn
%O A366234 0,3
%A A366234 _Paul D. Hanna_, Oct 05 2023