OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = Sum_{n>=0} n^(2*n) * exp(-n^2*x*exp(x)) * x^n * exp(n*x) / n!.
(2) A(x) = Sum_{n>=0} Stirling2(2*n,n) * x^n * exp(n*x).
(3) A( LambertW(x) ) = Sum_{n>=0} A007820(n)*x^n, where A007820(n) = A008277(2*n,n) for n >= 1 and A008277 equals the Stirling numbers of second kind.
(4) a(n) = n! * Sum_{k=0..n} Stirling2(2*k,k) * k^(n-k)/(n-k)!, for n >= 0.
a(n) ~ exp(w*(2-w)/4) * 2^(2*n) * n^(2*n) / (sqrt(1-w) * exp(2*n) * w^n * (2-w)^n), where w = -LambertW(-2*exp(-2)) = -A226775. - Vaclav Kotesovec, Apr 28 2026
EXAMPLE
E.g.f.: A(x) = 1 + x + 16*x^2/2! + 627*x^3/3! + 47644*x^4/4! + 5969205*x^5/5! + 1116212166*x^6/6! + 291418868755*x^7/7! + ...
where
A(x) = 1 + exp(-x*exp(x))*x*exp(x) + 2^4*exp(-2^2*x*exp(x))*x^2*exp(2*x)/2! + 3^6*exp(-3^2*x*exp(x))*x^3*exp(3*x)/3! + 4^8*exp(-4^2*x*exp(x))*x^4*exp(4*x)/4! + ... + n^(2*n)*exp(-n^2*x*exp(x))*x^n*exp(n*x)/n! + ...
Also, we have the integer series
A( LambertW(x) ) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 + 1323652*x^6 + ... + A007820(n)*x^n + ...
MATHEMATICA
Join[{1}, Table[n! * Sum[StirlingS2[2*k, k] * k^(n-k)/(n-k)!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Apr 28 2026 *)
PROG
(PARI) \\ Formula (1)
{a(n) = my(A=1, E = x*exp(x + x*O(x^n)));
A = 1 + sum(m=1, n, m^(2*m) * exp(-m^2*E) * E^m/m!); polcoef(EGF=A, n)}
{upto(n) = a(n); Vec(serlaplace(EGF))}
upto(20)
(PARI) \\ Formula (2)
{a(n) = my(A=1); A = sum(m=0, n, stirling(2*m, m, 2) * x^m * exp(m*x +x*O(x^n))); polcoef(EGF=A, n)}
{upto(n) = a(n); Vec(serlaplace(EGF))}
upto(20)
(PARI) \\ Formula (4)
{a(n) = n! * if(n==0, 1, sum(k=0, n, stirling(2*k, k, 2) * k^(n-k)/(n-k)! ) )}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 28 2026
STATUS
approved