A230323 E.g.f.: Sum_{n>=0} x^n/n! * LambertW(-n*x)/(-n*x).

1, 1, 3, 16, 149, 2196, 47167, 1380394, 52206345, 2457554248, 140408870651, 9563233300974, 765251809488157, 71012512562009500, 7552175961721086711, 911014865916673379026, 123562591612443767093393, 18704896422725902820936976, 3140424504257773679216307955

Offset: 0

Author

Paul D. Hanna, Oct 16 2013

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 149*x^4/4! + 2196*x^5/5! +...
Let W(x) = LambertW(-x)/(-x), then
W(n*x) = Sum_{k>=0} n^k*(k+1)^(k-1)*x^k/k! and
W(n*x) = [ Sum_{k>=0} (n*k+1)^(k-1)*x^k/k! ]^n
where
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! +...
Related expansions:
W(1*x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! +...
W(2*x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2000*x^4/4! + 41472*x^5/5! +...
W(3*x) = 1 + 3*x + 27*x^2/2! + 432*x^3/3! + 10125*x^4/4! + 314928*x^5/5! +...
W(4*x) = 1 + 4*x + 48*x^2/2! + 1024*x^3/3! + 32000*x^4/4! + 1327104*x^5/5! +...
W(5*x) = 1 + 5*x + 75*x^2/2! + 2000*x^3/3! + 78125*x^4/4! + 4050000*x^5/5! +...
...
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
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
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
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
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
...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A230321, A125281.

Programs

Mathematica

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 *)

PARI

{a(n)=sum(k=0,n,binomial(n,k)*(n-k)^k*(k+1)^(k-1))}
for(n=0,20,print1(a(n),", "))

PARI

{a(n)=local(A=1);
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)) );
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n)=local(A=1);
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 );
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n)=local(W=sum(m=0,n,(m+1)^(m-1)*x^m/m!)+x*O(x^n),A=1);
A=sum(m=0,n,x^m/m!*subst(W,x,m*x));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

PARI

{a(n)=local(LambertW=serreverse(x*exp(x+x*O(x^n))),A=1);
A=1+sum(m=1,n,x^m/m!*subst(LambertW,x,-m*x)/(-m*x));
n!*polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * (n-k)^k * (k+1)^(k-1).

E.g.f.: Sum_{n>=0} x^n/n! * Sum_{k>=0} n^k*(k+1)^(k-1)*x^k/k!.

E.g.f.: Sum_{n>=0} x^n/n! * [ Sum_{k>=0} (n*k+1)^(k-1)*x^k/k! ]^n.