A221160 G.f.: Sum_{n>=0} (4*n+1)^n * x^n / (1 + (4*n+1)*x)^n.
1, 5, 56, 864, 16896, 399360, 11059200, 350945280, 12551454720, 499415777280, 21879167385600, 1046394961920000, 54245114825932800, 3029690116944691200, 181363518724689100800, 11583863454028529664000, 786298610212845649920000, 56523637237014847291392000
Offset: 0
Keywords
Examples
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) + 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 +...
Programs
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Mathematica
a[n_] := (2*n + 3)*4^(n - 1)*n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Dec 23 2022 *)
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PARI
{a(n)=polcoeff(sum(m=0,n,((4*m+1)*x)^m/(1+(4*m+1)*x +x*O(x^n))^m), n)} for(n=0, 20, print1(a(n), ", "))
Formula
a(n) = (2*n+3) * 4^(n-1) * n! for n>0 with a(0)=1.
E.g.f.: (1 - 3*x + 4*x^2) / (1-4*x)^2.
From Amiram Eldar, Dec 23 2022: (Start)
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.
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)