OFFSET
0,2
COMMENTS
Conjecture: for n >= 6, a(n) (mod 6) equals [1, 0, 3, 0, 3, 2] 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 = 2 with p = LambertW(x)/x, r = x.
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.
(2) A(x) = Sum_{n>=0} 2^(n^2) * exp( LambertW(x) * 2^n ) * x^n / n!.
(3) A(x) = Sum_{n>=0} 2^(n^2) * (x/LambertW(x))^(2^n) * x^n / n!.
(4) A(x) = Sum_{n>=0} 2^(n*(n+1)) * x^n/n! * Sum_{k>=0} (2^n - k)^(k-1) * x^k/k!.
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).
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Aug 23 2025
EXAMPLE
E.g.f.: A(x) = 1 + 3*x + 23*x^2/2! + 708*x^3/3! + 82677*x^4/4! + 39043808*x^5/5! + 75384175459*x^6/6! + 594418947869568*x^7/7! + ...
where A(x) = Sum_{n>=0} (2^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!.
MATHEMATICA
nmax = 15; Join[{1}, Rest[CoefficientList[Series[Sum[(2^k*x + LambertW[x])^k/k!, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!]] (* Vaclav Kotesovec, Aug 23 2025 *)
PROG
(PARI) {a(n) = sum(k=0, n, binomial(n, k) * 2^(k*(k+1)) * (2^k - (n-k))^(n-k-1) )}
for(n=0, 12, print1(a(n), ", "))
(PARI) {a(n) = my(A = sum(m=0, n, (2^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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 23 2025
STATUS
approved