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%I #11 Dec 23 2022 07:40:18
%S 1,7,72,1056,19968,460800,12533760,392232960,13872660480,546979184640,
%T 23781703680000,1130106558873600,58263271479705600,
%U 3238634262940876800,193064390900475494400,12285915784575713280000,831229959367865401344000,59578968979556190388224000
%N G.f.: Sum_{n>=0} (4*n+3)^n * x^n / (1 + (4*n+3)*x)^n.
%F a(n) = (2*n+5) * 4^(n-1) * n! for n>0 with a(0)=1.
%F E.g.f.: (1 - x - 4*x^2) / (1-4*x)^2.
%F From _Amiram Eldar_, Dec 23 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 48*sqrt(Pi)*erfi(1/2) - 40*exp(1/4) + 1/5, where erfi is the imaginary error function.
%F Sum_{n>=0} (-1)^n/a(n) = 48*sqrt(Pi)*erf(1/2) - 56/exp(1/4) + 1/5, where erf is the error function. (End)
%e G.f.: A(x) = 1 + 7*x + 72*x^2 + 1056*x^3 + 19968*x^4 + 460800*x^5 +...
%e where
%e A(x) = 1 + 7*x/(1+7*x) + 11^2*x^2/(1+11*x)^2 + 15^3*x^3/(1+15*x)^3 + 19^4*x^4/(1+19*x)^4 + 23^5*x^5/(1+23*x)^5 +...
%t a[n_] := (2*n + 5)*4^(n - 1)*n!; a[0] = 1; Array[a, 20, 0] (* _Amiram Eldar_, Dec 23 2022 *)
%o (PARI) {a(n)=polcoeff(sum(m=0,n,((4*m+3)*x)^m/(1+(4*m+3)*x +x*O(x^n))^m), n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A187735, A014479, A187738, A187739, A221160, A187740, A333419.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 03 2013
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