OFFSET
0,3
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
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.
EXAMPLE
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
...
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 *)
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 16 2013
STATUS
approved