A217912 O.g.f.: Sum_{n>=0} 2*n^n * (3*n+2)^(n-1) * exp(-n*(3*n+2)*x) * x^n / n!.

1, 2, 22, 602, 26656, 1643054, 130318966, 12666846728, 1459524093232, 194626267782398, 29495119281572770, 5008297010070635978, 942044179147597185544, 194462342099815302424136, 43711609296992502659474632, 10628894996508864880841838416, 2780041837527932453797746700384

Offset

0,2

Comments

Compare the g.f. to the LambertW identity:

1 = Sum_{n>=0} 2*(3*n+2)^(n-1) * exp(-(3*n+2)*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} 2*(-1)^(n-k)*binomial(n,k) * k^n * (3*k+2)^(n-1).

a(n) = 1/n! * [x^n] Sum_{k>=0} 2*k^k*(3*k+2)^(k-1)*x^k / (1 + k*(3*k+2)*x)^(k+1).

a(n) = [x^n] 1 + x*(1+2*x)^(n-1) / Product_{k=1..n} (1 - 3*k*x).

a(n) = [x^n] 1 + x*(1-2*x)^(n-1) / Product_{k=1..n} (1 - (3*k+2)*x).

a(n) ~ 2^(2*n-1/6) * 3^(n-1) * n^(n-3/2) / (sqrt(Pi*(1-c)) * exp(n) * (2-c)^(n-1) * c^(n+1/3)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - Vaclav Kotesovec, May 22 2014

Example

O.g.f.: A(x) = 1 + 2*x + 22*x^2 + 602*x^3 + 26656*x^4 + 1643054*x^5 + ...
where
A(x) = 1 + 2*1^1*5^0*x*exp(-1*5*x) + 2*2^2*8^1*exp(-2*8*x)*x^2/2! + 2*3^3*11^2*exp(-3*11*x)*x^3/3! + 2*4^4*14^3*exp(-4*14*x)*x^4/4! + 2*5^5*17^4*exp(-5*17*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*2^(n-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, 2*k^k*(3*k+2)^(k-1)*x^k*exp(-k*(3*k+2)*x+x*O(x^n))/k!), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=1/n!*polcoeff(sum(k=0, n, 2*k^k*(3*k+2)^(k-1)*x^k/(1+k*(3*k+2)*x +x*O(x^n))^(k+1)), n)}
(PARI) {a(n)=1/n!*sum(k=0, n, 2*(-1)^(n-k)*binomial(n, k)*k^n*(3*k+2)^(n-1))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(1+2*x*(1+2*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+2*x*(1-2*x)^n/prod(k=0, n, 1-(3*k+2)*x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(Maxima) makelist( if n=0 then 1 else 1/n! * sum(2*(-1)^(n-k)*binomial(n, k) * k^n * (3*k+2)^(n-1), k, 0, n), n, 0, 30); /* Martin Ettl, Oct 15 2012 */

Crossrefs

Cf. A217900, A217910, A217911, A217913.

Sequence in context: A246740 A248798 A120419 * A210657 A177042 A308535

Adjacent sequences: A217909 A217910 A217911 * A217913 A217914 A217915

Keyword

nonn

Author

Paul D. Hanna, Oct 14 2012

Status

approved