A326087 E.g.f.: Sum_{n>=0} 3^(n^2) * W(x)^(3^n) * x^n/n!, where W(x) = LambertW(-x)/(-x).

1, 4, 102, 22021, 45222020, 864877715829, 151334332726346446, 240066322326063736396471, 3437872841642494470466400291880, 443629285085754349476473870485553832265, 515464807021418300124944982481178163507496409210, 5391365666992440498072962035380754373348405360720038284003, 507540246234432603646702271237262933001898870072439994003070887964860

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 = 3 and p = LambertW(-x)/(-x).

Links

Table of n, a(n) for n=0..12.

Formula

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

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

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

(3) Sum_{n>=0} 3^(n^2) * W(x)^(3^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) * 3^(k*(k+1)) * (3^k + n-k)^(n-k-1).

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

Example

E.g.f.: A(x) = 1 + 4*x + 102*x^2/2! + 22021*x^3/3! + 45222020*x^4/4! + 864877715829*x^5/5! + 151334332726346446*x^6/6! + 240066322326063736396471*x^7/7! + ...
such that
A(x) = W(x) + 3*W(x)^3*x + 3^4*W(x)^9*x^2/2! + 3^9*W(x)^27*x^3/3! + 3^16*W(x)^81*x^4/4! + 3^25*W(x)^243*x^5/5! + 3^36*W(x)^729*x^6/6! + ...
also
A(x) = 1 + (3 + W(x))*x + (3^2 + W(x))^2*x^2/2! + (3^3 + W(x))^3*x^3/3! + (3^4 + W(x))^4*x^4/4! + (3^5 + W(x))^5*x^5/5! + (3^6 + 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) * 3^(k*(k+1)) * (3^k + n-k)^(n-k-1) )}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=0} (3^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, (3^m + W)^m * x^m/m!), n)}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=0} 3^(n^2) * W(x)^(3^n) * x^n/n! */
{a(n) = my(W=serreverse(x/exp(x +x^2*O(x^n)))/x);
n!*polcoeff( sum(m=0, n, 3^(m^2) * W^(3^m) * x^m/m!), n)}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A326086.

Sequence in context: A220776 A180818 A211047 * A091330 A353312 A024056

Adjacent sequences: A326084 A326085 A326086 * A326088 A326089 A326090

Keyword

nonn

Author

Paul D. Hanna, Jun 11 2019

Status

approved