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E.g.f.: Sum_{n>=0} 3^n*(exp(n*x) - 1)^n.
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%I #18 Nov 12 2024 14:31:38

%S 1,3,75,4809,578415,112024353,31851411375,12493267169169,

%T 6464106627329055,4265281191267407073,3495556570494504442575,

%U 3483310917470882398369329,4147647341931988462919773695,5815857702618060221437908948993,9485411994735540168549266106329775

%N E.g.f.: Sum_{n>=0} 3^n*(exp(n*x) - 1)^n.

%H Vaclav Kotesovec, <a href="/A195263/b195263.txt">Table of n, a(n) for n = 0..210</a>

%F a(n) = Sum_{k=0..n} 3^k*k^n*k!*Stirling2(n,k).

%F a(n) ~ c * (1 + 3*exp(1/r))^n * r^(2*n) * n!^2 / sqrt(n), where r = 0.947093169766093813913446822751643203941993193936... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/3 and c = 0.36805443792839553744923868309093616341812244322234916... - _Vaclav Kotesovec_, Oct 04 2020

%e E.g.f.: A(x) = 1 + 3*x + 75*x^2/2! + 4809*x^3/3! + 578415*x^4/4! +...

%e where

%e A(x) = 1 + 3*(exp(x)-1) + 3^2*(exp(2*x)-1)^2 + 3^3*(exp(3*x)-1)^3 +...

%t Flatten[{1, Table[Sum[3^k * k^n * k! * StirlingS2[n,k], {k,0,n}], {n,1,20}]}] (* _Vaclav Kotesovec_, Oct 04 2020 *)

%o (PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(sum(m=0, n, 3^m*(exp(m*X)-1)^m), n)}

%o (PARI) {Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}

%o {a(n)=sum(k=0, n, 3^k*k^n*k!*Stirling2(n, k))}

%Y Cf. A195005, A122399, A301582, A338040.

%K nonn,changed

%O 0,2

%A _Paul D. Hanna_, Sep 13 2011