OFFSET
0,3
COMMENTS
Compare the g.f. to the LambertW identity:
1 = Sum_{n>=0} (3*n+1)^(n-1) * exp(-(3*n+1)*x) * x^n/n!.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..300
FORMULA
a(n) = 1/n! * Sum_{k=0..n} (-1)^(n-k)*binomial(n,k) * k^n * (3*k+1)^(n-1).
a(n) = 1/n! * [x^n] Sum_{k>=0} k^k*(3*k+1)^(k-1)*x^k / (1 + k*(3*k+1)*x)^(k+1).
a(n) = [x^n] 1 + x*(1+x)^(n-1) / Product_{k=1..n} (1 - 3*k*x).
a(n) = [x^n] 1 + x*(1-x)^(n-1) / Product_{k=1..n} (1 - (3*k+1)*x).
a(n) ~ 2^(2*n-4/3) * 3^(n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/6)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014
EXAMPLE
O.g.f.: A(x) = 1 + x + 10*x^2 + 262*x^3 + 11296*x^4 + 684172*x^5 +...
where
A(x) = 1 + 1^1*4^0*x*exp(-1*4*x) + 2^2*7^1*exp(-2*7*x)*x^2/2! + 3^3*10^2*exp(-3*10*x)*x^3/3! + 4^4*13^3*exp(-4*13*x)*x^4/4! + 5^5*16^4*exp(-5*16*x)*x^5/5! +...
simplifies to a power series in x with integer coefficients.
MATHEMATICA
Flatten[{1, Table[Sum[Binomial[n-1, j]*3^j*StirlingS2[n+j, n], {j, 0, n-1}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 22 2014 *)
PROG
(PARI) {a(n)=polcoeff(sum(k=0, n, k^k*(3*k+1)^(k-1)*x^k*exp(-k*(3*k+1)*x+x*O(x^n))/k!), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, k^k*(3*k+1)^(k-1)*x^k/(1+k*(3*k+1)*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=1/n!*sum(k=0, n, (-1)^(n-k)*binomial(n, k)*k^n*(3*k+1)^(n-1))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(1+x*(1+x)^(n-1)/prod(k=0, n, 1-3*k*x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(1+x*(1-x)^n/prod(k=0, n, 1-(3*k+1)*x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2012
STATUS
approved