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%I #12 Jan 03 2013 01:50:47
%S 1,4,33,378,5508,97200,2012040,47764080,1278607680,38093690880,
%T 1249949232000,44783895340800,1739500776921600,72804471541401600,
%U 3266273336880153600,156364149105964800000,7955807906511489024000,428712969452770050048000,24390705726366524633088000
%N G.f.: Sum_{n>=0} (3*n+1)^n * x^n / (1 + (3*n+1)*x)^n.
%C More generally,
%C if Sum_{n>=0} a(n)*x^n = Sum_{n>=0} (b*n+c)^n * x^n / (1 + (b*n+c)*x)^n,
%C 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)
%C so that a(n) = (b*n + (b+2*c)) * b^(n-1) * n!/2 for n>0 with a(0)=1.
%F a(n) = (3*n+5) * 3^(n-1) * n!/2 for n>0 with a(0)=1.
%F E.g.f.: (2 - 4*x + 3*x^2) / (2*(1-3*x)^2).
%e G.f.: A(x) = 1 + 4*x + 33*x^2 + 378*x^3 + 5508*x^4 + 97200*x^5 +...
%e where
%e A(x) = 1 + 4*x/(1+4*x) + 7^2*x^2/(1+7*x)^2 + 10^3*x^3/(1+10*x)^3 + 13^4*x^4/(1+13*x)^4 + 16^5*x^5/(1+16*x)^5 +...
%o (PARI) {a(n)=polcoeff(sum(m=0,n,((3*m+1)*x)^m/(1+(3*m+1)*x +x*O(x^n))^m), n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A187735, A014479, A187739, A221160, A221161, A187740.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 03 2013
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