A187738 G.f.: Sum_{n>=0} (3*n+1)^n * x^n / (1 + (3*n+1)*x)^n.

1, 4, 33, 378, 5508, 97200, 2012040, 47764080, 1278607680, 38093690880, 1249949232000, 44783895340800, 1739500776921600, 72804471541401600, 3266273336880153600, 156364149105964800000, 7955807906511489024000, 428712969452770050048000, 24390705726366524633088000

Offset: 0

Author

Paul D. Hanna, Jan 03 2013

Keywords

• nonn

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.

Examples

			G.f.: A(x) = 1 + 4*x + 33*x^2 + 378*x^3 + 5508*x^4 + 97200*x^5 +...
where
A(x) = 1 + 4*x/(1+4*x) + 7^2*x^2/(1+7*x)^2 + 10^3*x^3/(1+10*x)^3 + 13^4*x^4/(1+13*x)^4 + 16^5*x^5/(1+16*x)^5 +...
		

Crossrefs

Cf. A187735, A014479, A187739, A221160, A221161, A187740.

Programs

PARI

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

Formula

a(n) = (3*n+5) * 3^(n-1) * n!/2 for n>0 with a(0)=1.

E.g.f.: (2 - 4*x + 3*x^2) / (2*(1-3*x)^2).