A229036 G.f.: Sum_{n>=0} (3*n-1)^n * x^n / (1 + (3*n-1)*x)^n.

1, 2, 21, 270, 4212, 77760, 1662120, 40415760, 1102248000, 33331979520, 1107097891200, 40069801094400, 1569793384051200, 66185883219456000, 2988292627358438400, 143855017177487616000, 7355369573944584192000, 398090614491857903616000, 22737098558477268725760000

Offset: 0

Author

Paul D. Hanna, Sep 11 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

			O.g.f.: A(x) = 1 + 2*x + 21*x^2 + 270*x^3 + 4212*x^4 + 77760*x^5 +...
where
A(x) = 1 + 2*x/(1+2*x) + 5^2*x^2/(1+5*x)^2 + 8^3*x^3/(1+8*x)^3 + 11^4*x^4/(1+11*x)^4 + 14^5*x^5/(1+14*x)^5 +...
E.g.f.: E(x) = 1 + 2*x + 21*x^2/2! + 270*x^3/3! + 4212*x^4/4! + 77760*x^5/5! +...
where
E(x) =  1 + 2*x + 21/2*x^2 + 45*x^3 + 351/2*x^4 + 648*x^5 + 4617/2*x^6 +...
which is the expansion of: (2 - 8*x + 15*x^2) / (2 - 12*x + 18*x^2).
		

Crossrefs

Cf. A229039, A187735, A187738, A187739, A221160, A221161, A187740.

Programs

Mathematica

Join[{1},Table[(3n+1)3^(n-1) n!/2,{n,20}]] (* Harvey P. Dale, Feb 10 2015 *)

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), ", "))

PARI

{a(n) = if(n==0,1,(3*n+1)*3^(n-1)*n!/2)}
for(n=0, 20, print1(a(n), ", "))

Formula

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

E.g.f.: (2 - 8*x + 15*x^2)/(2*(1-3*x)^2).