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

1, 1, 10, 126, 2392, 60600, 1916304, 72917488, 3246171520, 165609099648, 9529240349440, 610657739172096, 43136025287678976, 3330356645773880320, 279024535906794539008, 25214258236430338160640, 2444656672390982922502144, 253144081975231633923342336

Offset

0,3

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.

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = n! * Sum{k=0..n} binomial(n+1, n-k)/(n+1) * 4^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) * 4^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(4*x*A(x)).

(2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(4*x)) ).

(3) A( x/(1 + x*exp(4*x)) ) = 1 + x*exp(4*x).

(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.

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

(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)).

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

Example

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! + ...
where A(x) satisfies A(x) = 1 + x*A(x) * exp(4*x*A(x))
also
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! + ...
and
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! + ...
RELATED SERIES.
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! + ...

Mathematica

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 *)

Prog

(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) * 4^k * (n-k)^k/k!)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A = (1/x) * serreverse( x/(1 + x*exp(4*x +O(x^(n+2)))) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A161633, A366232, A366233, A366235.

Cf. A365774 (dual), A245265 (exp(x*A(x)).

Sequence in context: A079609 A101599 A002458 * A192600 A079241 A270965

Adjacent sequences: A366231 A366232 A366233 * A366235 A366236 A366237

Keyword

nonn

Author

Paul D. Hanna, Oct 05 2023

Status

approved