A386658 E.g.f.: Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.

1, 6, 674, 2000229, 153566609748, 298500361403750381, 14557504055095871311168750, 17765160070810827062009088144577731, 542112188572462226990932242595876785196798632, 413592212104548192173492724488185195719396124921931347641

Offset: 0

Author

Paul D. Hanna, Aug 23 2025

Keywords

• nonn
• new

Comments

Conjecture: for n >= 6, a(n) (mod 6) equals [4, 3, 0, 3, 0, 5] repeating.

In general, the following sums are equal:

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

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

here, q = 5 with p = LambertW(x)/x, r = x.

Examples

			E.g.f.: A(x) = 1 + 6*x + 674*x^2/2! + 2000229*x^3/3! + 153566609748*x^4/4! + 298500361403750381*x^5/5! + 14557504055095871311168750*x^6/6! + ...
where A(x) = Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.
RELATED SERIES.
LambertW(x) = x - 2*x^2/2! + 3^2*x^3/3! - 4^3*x^4/4! + 5^4*x^5/5! - 6^5*x^6/6! + 7^6*x^7/7! + ... + (-1)^(n-1) * n^(n-1)*x^n/n! + ...
where exp(LambertW(x)) = x/LambertW(x);
also, (x/LambertW(x))^y = Sum_{k>=0} y*(y - k)^(k-1) * x^k/k!.
		

Crossrefs

Cf. A386655 (q=2), A386656 (q=3), A386657 (q=4), A386648.

Programs

PARI

{a(n,q=5) = sum(k=0,n, binomial(n,k) * q^(k*(k+1)) * (q^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))

PARI

{a(n,q=5) = my(A = sum(m=0, n, (q^m + lambertw(x +x^3*O(x^n))/x)^m *x^m/m! )+x*O(x^n)); n! * polcoeff(A, n)}
for(n=0, 12, print1(a(n), ", "))

Formula

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

(1) A(x) = Sum_{n>=0} (5^n*x + LambertW(x))^n / n!.

(2) A(x) = Sum_{n>=0} 5^(n^2) * exp( LambertW(x) * 5^n ) * x^n / n!.

(3) A(x) = Sum_{n>=0} 5^(n^2) * (x/LambertW(x))^(5^n) * x^n / n!.

(4) A(x) = Sum_{n>=0} 5^(n*(n+1)) * x^n/n! * Sum_{k>=0} (5^n - k)^(k-1) * x^k/k!.

a(n) = Sum_{k=0..n} binomial(n,k) * 5^(k*(k+1)) * (5^k - (n-k))^(n-k-1).

a(n) = Sum_{k=0..n} binomial(n,k) * 5^(n*k) * (1 - (n-k)/5^k)^(n-k-1).