This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A230323 #31 Mar 19 2024 09:35:15
%S A230323 1,1,3,16,149,2196,47167,1380394,52206345,2457554248,140408870651,
%T A230323 9563233300974,765251809488157,71012512562009500,7552175961721086711,
%U A230323 911014865916673379026,123562591612443767093393,18704896422725902820936976,3140424504257773679216307955
%N A230323 E.g.f.: Sum_{n>=0} x^n/n! * LambertW(-n*x)/(-n*x).
%H A230323 Vincenzo Librandi, <a href="/A230323/b230323.txt">Table of n, a(n) for n = 0..200</a>
%F A230323 a(n) = Sum_{k=0..n} binomial(n,k) * (n-k)^k * (k+1)^(k-1).
%F A230323 E.g.f.: Sum_{n>=0} x^n/n! * Sum_{k>=0} n^k*(k+1)^(k-1)*x^k/k!.
%F A230323 E.g.f.: Sum_{n>=0} x^n/n! * [ Sum_{k>=0} (n*k+1)^(k-1)*x^k/k! ]^n.
%e A230323 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2196*x^5/5! +...
%e A230323 Let W(x) = LambertW(-x)/(-x), then
%e A230323 W(n*x) = Sum_{k>=0} n^k*(k+1)^(k-1)*x^k/k! and
%e A230323 W(n*x) = [ Sum_{k>=0} (n*k+1)^(k-1)*x^k/k! ]^n
%e A230323 where
%e A230323 A(x) = 1 + x*W(x) + x^2*W(2*x)/2! + x^3*W(3*x)/3! + x^4*W(4*x)/4! + x^5*W(5*x)/5! + x^6*W(6*x)/6! +...
%e A230323 Related expansions:
%e A230323 W(1*x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...
%e A230323 W(2*x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + 41472*x^5/5! +...
%e A230323 W(3*x) = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + 314928*x^5/5! +...
%e A230323 W(4*x) = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + 1327104*x^5/5! +...
%e A230323 W(5*x) = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + 4050000*x^5/5! +...
%e A230323 ...
%e A230323 W(1*x) = (1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! +...)^1
%e A230323 W(2*x) = (1 + x + 5*x^2/2! + 7^2*x^3/3! + 9^3*x^4/4! + 11^4*x^5/5! +...)^2
%e A230323 W(3*x) = (1 + x + 7*x^2/2! + 10^2*x^3/3! + 13^3*x^4/4! + 16^4*x^5/5! +...)^3
%e A230323 W(4*x) = (1 + x + 9*x^2/2! + 13^2*x^3/3! + 17^3*x^4/4! + 21^4*x^5/5! +...)^4
%e A230323 W(5*x) = (1 + x + 11*x^2/2! + 16^2*x^3/3! + 21^3*x^4/4! + 26^4*x^5/5! +...)^5
%e A230323 ...
%t A230323 Flatten[{1,Table[Sum[Binomial[n,k] * (n-k)^k * (k+1)^(k-1),{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jul 29 2014 *)
%o A230323 (PARI) {a(n)=sum(k=0,n,binomial(n,k)*(n-k)^k*(k+1)^(k-1))}
%o A230323 for(n=0,20,print1(a(n),", "))
%o A230323 (PARI) {a(n)=local(A=1);
%o A230323 A=sum(m=0,n,x^m/m!*sum(j=0,n,m^j*(j+1)^(j-1)*x^j/j! +x*O(x^n)) );
%o A230323 n!*polcoeff(A,n)}
%o A230323 for(n=0,20,print1(a(n),", "))
%o A230323 (PARI) {a(n)=local(A=1);
%o A230323 A=sum(m=0,n,x^m/m!*sum(j=0,n,(m*j+1)^(j-1)*x^j/j! +x*O(x^n))^m );
%o A230323 n!*polcoeff(A,n)}
%o A230323 for(n=0,20,print1(a(n),", "))
%o A230323 (PARI) {a(n)=local(W=sum(m=0,n,(m+1)^(m-1)*x^m/m!)+x*O(x^n),A=1);
%o A230323 A=sum(m=0,n,x^m/m!*subst(W,x,m*x));
%o A230323 n!*polcoeff(A,n)}
%o A230323 for(n=0,20,print1(a(n),", "))
%o A230323 (PARI) {a(n)=local(LambertW=serreverse(x*exp(x+x*O(x^n))),A=1);
%o A230323 A=1+sum(m=1,n,x^m/m!*subst(LambertW,x,-m*x)/(-m*x));
%o A230323 n!*polcoeff(A,n)}
%o A230323 for(n=0,20,print1(a(n),", "))
%Y A230323 Cf. A230321, A125281.
%K A230323 nonn
%O A230323 0,3
%A A230323 _Paul D. Hanna_, Oct 16 2013