A187739 G.f.: Sum_{n>=0} (3*n+2)^n * x^n / (1 + (3*n+2)*x)^n.

1, 5, 39, 432, 6156, 106920, 2187000, 51438240, 1366787520, 40474546560, 1321374902400, 47140942464000, 1824354473356800, 76113765702374400, 3405263691641011200, 162618715070203392000, 8256027072794941440000, 444024146933226123264000, 25217509310311152586752000

Offset

0,2

Comments

More generally,

if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,

then Sum_{n>=0} a(n)*x^n/n! = (2 - 2*(b-c)*x + b*(b-2*c)*x^2)/(2*(1-b*x)^2)

so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.

Links

Table of n, a(n) for n=0..18.

Formula

a(n) = (3*n+7) * 3^(n-1) * n!/2 for n>0 with a(0)=1.

E.g.f.: (2 - 2*x - 3*x^2) / (2*(1-3*x)^2).

Example

G.f.: A(x) = 1 + 5*x + 39*x^2 + 432*x^3 + 6156*x^4 + 106920*x^5 +...
where
A(x) = 1 + 5*x/(1+5*x) + 8^2*x^2/(1+8*x)^2 + 11^3*x^3/(1+11*x)^3 + 14^4*x^4/(1+14*x)^4 + 17^5*x^5/(1+17*x)^5 +...

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, ((3*m+2)*x)^m/(1+(3*m+2)*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A187735, A014479, A187738, A221160, A221161, A187740.

Sequence in context: A127189 A121354 A122486 * A067083 A199244 A322884

Adjacent sequences: A187736 A187737 A187738 * A187740 A187741 A187742

Keyword

nonn

Author

Paul D. Hanna, Jan 03 2013

Status

approved