A393877 - OEIS

%I #12 Apr 28 2026 08:46:17

%S 1,1,16,627,47644,5969205,1116212166,291418868755,101275736219176,

%T 45202818179409417,25200537485351137930,17162078056673742593511,

%U 14019641172860291811663228,13530737107158473563692308029,15231714898939447123185454990702,19780681881247619480407404640210395

%N Expansion of e.g.f. Sum_{n>=0} Stirling2(2*n,n) * x^n * exp(n*x).

%H Paul D. Hanna, <a href="/A393877/b393877.txt">Table of n, a(n) for n = 0..300</a>

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

%F (1) A(x) = Sum_{n>=0} n^(2*n) * exp(-n^2*x*exp(x)) * x^n * exp(n*x) / n!.

%F (2) A(x) = Sum_{n>=0} Stirling2(2*n,n) * x^n * exp(n*x).

%F (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.

%F (4) a(n) = n! * Sum_{k=0..n} Stirling2(2*k,k) * k^(n-k)/(n-k)!, for n >= 0.

%F 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

%e 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! + ...

%e where

%e 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! + ...

%e Also, we have the integer series

%e A( LambertW(x) ) = 1 + x + 7*x^2 + 90*x^3 + 1701*x^4 + 42525*x^5 + 1323652*x^6 + ... + A007820(n)*x^n + ...

%t 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 *)

%o (PARI) \\ Formula (1)

%o {a(n) = my(A=1, E = x*exp(x + x*O(x^n)));

%o A = 1 + sum(m=1, n, m^(2*m) * exp(-m^2*E) * E^m/m!); polcoef(EGF=A, n)}

%o {upto(n) = a(n); Vec(serlaplace(EGF))}

%o upto(20)

%o (PARI) \\ Formula (2)

%o {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)}

%o {upto(n) = a(n); Vec(serlaplace(EGF))}

%o upto(20)

%o (PARI) \\ Formula (4)

%o {a(n) = n! * if(n==0, 1, sum(k=0, n, stirling(2*k, k, 2) * k^(n-k)/(n-k)! ) )}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A006153, A393878, A395122, A007820, A008277, A000108.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 28 2026