cp's OEIS Frontend

A187746 G.f.: Sum_{n>=0} (2*n+x)^n * x^n / (1 + 2*n*x + x^2)^n.

%I A187746 #8 Jan 03 2013 10:25:23
%S A187746 1,2,13,100,984,11712,163200,2603520,46771200,934133760,20530298880,
%T A187746 492355584000,12793813401600,358063276032000,10737974299852800,
%U A187746 343513154086502400,11676590580695040000,420271561157640192000,15967576932074127360000
%N A187746 G.f.: Sum_{n>=0} (2*n+x)^n * x^n / (1 + 2*n*x + x^2)^n.
%F A187746 a(n) = (2*n^2+2*n+1) * 2^(n-2) * (n-1)! for n>1 with a(0)=1, a(1)=2.
%F A187746 E.g.f.: 1/2 + 1/(2*(1-2*x)^2) - x/2 - log(1-2*x)/4.
%F A187746 E.g.f.: Sum_{n>=0} a(n+1)*x^n/n! = 2/(1-2*x)^3 + x/(1-2*x).
%e A187746 G.f.: A(x) = 1 + 2*x + 13*x^2 + 100*x^3 + 984*x^4 + 11712*x^5 +...
%e A187746 where
%e A187746 A(x) = 1 + (2+x)*x/(1+2*x+x^2) + (4+x)^2*x^2/(1+4*x+x^2)^2 + (6+x)^3*x^3/(1+6*x+x^2)^3 + (8+x)^4*x^4/(1+8*x+x^2)^4 + (10+x)^5*x^5/(1+10*x+x^2)^5 +...
%o A187746 (PARI) {a(n)=polcoeff( sum(m=0, n, (2*m+x)^m*x^m/(1+2*m*x+x^2 +x*O(x^n))^m), n)}
%o A187746 for(n=0, 30, print1(a(n), ", "))
%o A187746 (PARI) {a(n)=if(n==0,1,if(n==1,2,(2*n^2+2*n+1)*2^(n-2)*(n-1)!))}
%o A187746 for(n=0, 30, print1(a(n), ", "))
%o A187746 (PARI) {a(n)=n!*polcoeff(1/2 + 1/(2*(1-2*x)^2) - x/2 - log(1-2*x +x*O(x^n))/4, n)}
%o A187746 for(n=0, 30, print1(a(n), ", "))
%Y A187746 Cf. A187742, A187735.
%K A187746 nonn
%O A187746 0,2
%A A187746 _Paul D. Hanna_, Jan 03 2013