cp's OEIS Frontend

A221160 G.f.: Sum_{n>=0} (4*n+1)^n * x^n / (1 + (4*n+1)*x)^n.

%I A221160 #12 Dec 23 2022 07:40:13
%S A221160 1,5,56,864,16896,399360,11059200,350945280,12551454720,499415777280,
%T A221160 21879167385600,1046394961920000,54245114825932800,
%U A221160 3029690116944691200,181363518724689100800,11583863454028529664000,786298610212845649920000,56523637237014847291392000
%N A221160 G.f.: Sum_{n>=0} (4*n+1)^n * x^n / (1 + (4*n+1)*x)^n.
%F A221160 a(n) = (2*n+3) * 4^(n-1) * n!  for n>0 with a(0)=1.
%F A221160 E.g.f.: (1 - 3*x + 4*x^2) / (1-4*x)^2.
%F A221160 From _Amiram Eldar_, Dec 23 2022: (Start)
%F A221160 Sum_{n>=0} 1/a(n) = 8*exp(1/4) - 1/3 - 8*sqrt(Pi)*erfi(1/2), where erfi is the imaginary error function.
%F A221160 Sum_{n>=0} (-1)^n/a(n) = 8*sqrt(Pi)*erf(1/2) - 8/exp(1/4) - 1/3, where erf is the error function. (End)
%e A221160 G.f.: A(x) = 1 + 5*x + 39*x^2 + 432*x^3 + 6156*x^4 + 106920*x^5 +...
%e A221160 where
%e A221160 A(x) = 1 + 5*x/(1+5*x) + 9^2*x^2/(1+9*x)^2 + 13^3*x^3/(1+13*x)^3 + 17^4*x^4/(1+17*x)^4 + 21^5*x^5/(1+21*x)^5 +...
%t A221160 a[n_] := (2*n + 3)*4^(n - 1)*n!; a[0] = 1; Array[a, 20, 0] (* _Amiram Eldar_, Dec 23 2022 *)
%o A221160 (PARI) {a(n)=polcoeff(sum(m=0,n,((4*m+1)*x)^m/(1+(4*m+1)*x +x*O(x^n))^m), n)}
%o A221160 for(n=0, 20, print1(a(n), ", "))
%Y A221160 Cf. A187735, A014479, A187738, A187739, A221161, A187740, A333419.
%K A221160 nonn
%O A221160 0,2
%A A221160 _Paul D. Hanna_, Jan 03 2013