OFFSET
1,3
COMMENTS
Conjecture: for n > 6, a(n) (mod 6) equals [2,3,4,3,2,5] 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 = A(x) with p = LambertW(-x), r = 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..102
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) 1 = Sum_{n>=0} ( A(x)^n + LambertW(-x) )^n / n!.
(2) 1 = Sum_{n>=0} A(x)^(n^2) * exp( LambertW(-x) * A(x)^n ) / n!.
(3) 1 = Sum_{n>=0} A(x)^(n^2) * (-x/LambertW(-x))^(A(x)^n) / n!.
(4) 1 = Sum_{n>=0} A(x)^(n*(n+1))/n! * Sum_{k>=0} (A(x)^n - k)^(k-1) * (-x)^k/k!.
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 10*x^3/3! + 63*x^4/4! + 806*x^5/5! + 9485*x^6/6! + 161540*x^7/7! + 2752155*x^8/8! + 59021506*x^9/9! + 1310350929*x^10/10! + ...
where 1 = Sum_{n>=0} ( A(x)^n + 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! + ... + 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!.
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=0, n, A=concat(A, 0);
A[#A] = polcoeff(1 - sum(m=0, #A, (Ser(A)^m + lambertw(-x +x^3*O(x^n)))^m /m! ), #A-1) ); n!*A[n+1]}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 12 2025
STATUS
approved