A366233 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*A(x) * exp(3*x*A(x)).

1, 1, 8, 87, 1428, 31125, 847818, 27785205, 1065267864, 46802921769, 2319200977230, 127985909702409, 7785440359004916, 517616528753919933, 37344834374921549154, 2906043724955696034285, 242627026212699695954352, 21634774261037677172406609, 2052077586846349144929739542

Offset: 0

Author

Paul D. Hanna, Oct 05 2023

Keywords

• nonn

Comments

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

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

(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)),

(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)),

(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)),

(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.

Examples

			E.g.f.: A(x) = 1 + x + 8*x^2/2! + 87*x^3/3! + 1428*x^4/4! + 31125*x^5/5! + 847818*x^6/6! + 27785205*x^7/7! + 1065267864*x^8/8! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(3*x*A(x))
also
A(x) = 1 + 1^0*x*A(x)*exp(+2*x*A(x))/1! + 2^1*x^2*A(x)^2*exp(+1*x*A(x))/2! + 3^2*x^3*A(x)^3*exp(-0*x*A(x))/3! + 4^3*x^4*A(x)^4*exp(-1*x*A(x))/4! + 5^4*x^5*A(x)^5*exp(-2*x*A(x))/5! + 6^5*x^6*A(x)^6*exp(-3*x*A(x))/6! + ...
and
A(x) = 1 + 4*1*4^(-1)*x*A(x)*exp(-1*x*A(x))/1! + 4*2*5^0*x^2*A(x)^2*exp(-2*x*A(x))/2! + 4*3*6^1*x^3*A(x)^3*exp(-3*x*A(x))/3! + 4*4*7^2*x^4*A(x)^4*exp(-4*x*A(x))/4! + 4*5*8^3*x^5*A(x)^5*exp(-5*x*A(x))/5! + ...
RELATED SERIES.
exp(x*A(x)) = 1 + x + 3*x^2/2! + 31*x^3/3! + 469*x^4/4! + 9681*x^5/5! + 254701*x^6/6! + ... + A212917(n)*x^n/n! + ...
		

Crossrefs

Cf. A161633, A366232, A366234, A366235.

Cf. A365773 (dual), A212917 (exp(x*A(x))).

Programs

Mathematica

nmax = 20; A[] = 0; Do[A[x] = 1 + x*A[x] * E^(3*x*A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* Vaclav Kotesovec, Oct 06 2023 *)

PARI

/* a(n,m) = coefficient of x^n/n! in A(x)^m, here at m = 1 */
{a(n, m=1) = n!*sum(k=0, n, binomial(n+m, n-k)*m/(n+m) * 3^k * (n-k)^k/k!)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(3*x +O(x^(n+2)))) )); n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+1, n-k)/(n+1) * 3^k * (n-k)^k / k!.

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) * 3^k * (n-k)^k/k!.

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

(1) A(x) = 1 + x*A(x) * exp(3*x*A(x)).

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

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

(4) A(x) = 1 + (m+1) * Sum_{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-3)*x*A(x)) for all fixed nonnegative m.

(4.a) A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n/n! * A(x)^n * exp(-(n-3)*x*A(x)).

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

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

(4.d) A(x) = 1 + 4 * Sum_{n>=1} n*(n+3)^(n-2) * x^n/n! * A(x)^n * exp(-n*x*A(x)).

(4.e) A(x) = 1 + 5 * Sum_{n>=1} n*(n+4)^(n-2) * x^n/n! * A(x)^n * exp(-(n+1)*x*A(x)).

a(n) ~ 3^n * (1 + 2*LambertW(sqrt(3)/2))^(n + 3/2) * n^(n-1) / (sqrt(1 + LambertW(sqrt(3)/2)) * 2^(2*n + 2) * exp(n) * LambertW(sqrt(3)/2)^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2023