A326086 - OEIS

A326086

E.g.f.: Sum_{n>=0} 2^(n^2) * W(x)^(2^n) * x^n/n!, where W(x) = LambertW(-x)/(-x).

1, 3, 27, 768, 84749, 39243888, 75455521471, 594516437142848, 19031406668609378713, 2460691831851012225530880, 1280084975976508309274383168595, 2672769856819326065224240364145984960, 22366167385647246803941382124073221290016997, 749477179363850141651152204676749367782390886880128, 100519481588580178976399215728025075471667058977184517953799, 53944581177412223136063119942558559584461301815948131410099383585344

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

More generally, the following sums are equal:

(1) Sum_{n>=0} (q^n + p)^n * x^n/n!,

(2) Sum_{n>=0} q^(n^2) * exp(p*q^n*x) * x^n/n!;

here, q = 2 and p = LambertW(-x)/(-x).

LINKS

Table of n, a(n) for n=0..15.

FORMULA

E.g.f. Sum_{n>=0} a(n)*x^n/n! may be defined by the following sums:

(1) Sum_{n>=0} (2^n + W(x))^n * x^n/n!,

(2) Sum_{n>=0} log(exp(2^n*x) * W(x))^n / n!,

(3) Sum_{n>=0} 2^(n^2) * W(x)^(2^n) * x^n/n!,

where W(x) = exp(x*W(x)) = LambertW(-x)/(-x).

FORMULAS FOR TERMS.

a(n) = Sum_{k=0..n} binomial(n,k) * 2^(k*(k+1)) * (2^k + n-k)^(n-k-1).

a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n*k) * (1 + (n-k)/2^k)^(n-k-1).

EXAMPLE

E.g.f.: A(x) = 1 + 3*x + 27*x^2/2! + 768*x^3/3! + 84749*x^4/4! + 39243888*x^5/5! + 75455521471*x^6/6! + 594516437142848*x^7/7! + 19031406668609378713*x^8/8! + ...

such that

A(x) = W(x) + 2*W(x)^2*x + 2^4*W(x)^4*x^2/2! + 2^9*W(x)^8*x^3/3! + 2^16*W(x)^16*x^4/4! + 2^25*W(x)^32*x^5/5! + 2^36*W(x)^64*x^6/6! + ...

also

A(x) = 1 + (2 + W(x))*x + (2^2 + W(x))^2*x^2/2! + (2^3 + W(x))^3*x^3/3! + (2^4 + W(x))^4*x^4/4! + (2^5 + W(x))^5*x^5/5! + (2^6 + W(x))^6*x^6/6! + ...

where W(x) begins

W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! + ... + (n+1)^(n-1)*x^n/n! + ...

so that W(x) = exp(x*W(x)) = LambertW(-x)/(-x).

PROG

(PARI) /* binomial formula for terms */

{a(n) = sum(k=0, n, binomial(n, k) * 2^(k*(k+1)) * (2^k + n-k)^(n-k-1) )}

for(n=0, 15, print1(a(n), ", "))

(PARI) /* E.g.f.: Sum_{n>=0} (2^n + W(x))^n * x^n/n! */

{a(n) = my(W = serreverse(x/exp(x +x^2*O(x^n)))/x);

n!*polcoeff( sum(m=0, n, (2^m + W)^m * x^m/m!), n)}

for(n=0, 15, print1(a(n), ", "))

(PARI) /* E.g.f.: Sum_{n>=0} 2^(n^2) * W(x)^(2^n) * x^n/n! */

{a(n) = my(W = serreverse(x/exp(x +x^2*O(x^n)))/x);

n!*polcoeff( sum(m=0, n, 2^(m^2) * W^(2^m) * x^m/m!), n)}