A366232 - OEIS

%I #20 Oct 08 2023 06:45:21

%S 1,1,6,54,728,13000,290352,7800016,245115264,8826560640,358463525120,

%T 16212238054144,808215885708288,44035925223746560,2603618739995621376,

%U 166031767704180111360,11359670347331723952128,830065763154102656204800,64518486557995327748898816

%N Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(2*x*A(x)).

%C Related identities which hold formally for all Maclaurin series F(x):

%C (1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n/n! * F(x)^n * exp(-n*x*F(x)),

%C (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 (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 (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 (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 a(n) = n! * Sum{k=0..n} binomial(n+1, n-k)/(n+1) * 2^k * (n-k)^k / k!.

%F 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) * 2^k * (n-k)^k/k!.

%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

%F (1) A(x) = 1 + x*A(x) * exp(2*x*A(x)).

%F (2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(2*x)) ).

%F (3) A( x/(1 + x*exp(2*x)) ) = 1 + x*exp(2*x).

%F (4) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-2)*x*A(x)) for all fixed nonnegative m.

%F (4.a) A(x) = 1 + Sum{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-2)*x*A(x)).

%F (4.b) A(x) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n/n! * A(x)^n * exp(-(n-1)*x*A(x)).

%F (4.c) A(x) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).

%F (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 a(n) ~ n^(n-1) * (1 + 2*LambertW(1/sqrt(2)))^(n + 3/2) / (sqrt(1 + LambertW(1/sqrt(2))) * 2^(n+2) * exp(n) * LambertW(1/sqrt(2))^(2*n + 3/2)). - _Vaclav Kotesovec_, Oct 06 2023

%e E.g.f.: A(x) = 1 + x + 6*x^2/2! + 54*x^3/3! + 728*x^4/4! + 13000*x^5/5! + 290352*x^6/6! + 7800016*x^7/7! + 245115264*x^8/8! + ...

%e where A(x) satisfies A(x) = 1 + x*A(x) * exp(2*x*A(x))

%e also

%e A(x) = 1 + 1^0*x*A(x)*exp(+1*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(-0*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(-2*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-3*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-4*x*A(x))/6! + ...

%e and

%e A(x) = 1 + 3*1*3^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 3*2*4^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 3*3*5^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 3*4*6^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 3*5*7^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...

%e RELATED SERIES.

%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 25*x^3/3! + 313*x^4/4! + 5341*x^5/5! + ... + A212722(n)*x^n/n! + ...

%t nmax = 20; A[_] = 0; Do[A[x_] = 1 + x*A[x] * E^(2*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* _Vaclav Kotesovec_, Oct 06 2023 *)

%o (PARI)  /* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */

%o {a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 2^k * (n-k)^k/k!)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) {a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(2*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A161633, A366233, A366234, A366235.

%Y Cf. A365772 (dual), A212722 (exp(x*A(x)).

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 05 2023