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%I #14 Jan 03 2013 01:51:04
%S 1,5,39,432,6156,106920,2187000,51438240,1366787520,40474546560,
%T 1321374902400,47140942464000,1824354473356800,76113765702374400,
%U 3405263691641011200,162618715070203392000,8256027072794941440000,444024146933226123264000,25217509310311152586752000
%N G.f.: Sum_{n>=0} (3*n+2)^n * x^n / (1 + (3*n+2)*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+7) * 3^(n-1) * n!/2 for n>0 with a(0)=1.
%F E.g.f.: (2 - 2*x - 3*x^2) / (2*(1-3*x)^2).
%e G.f.: A(x) = 1 + 5*x + 39*x^2 + 432*x^3 + 6156*x^4 + 106920*x^5 +...
%e where
%e A(x) = 1 + 5*x/(1+5*x) + 8^2*x^2/(1+8*x)^2 + 11^3*x^3/(1+11*x)^3 + 14^4*x^4/(1+14*x)^4 + 17^5*x^5/(1+17*x)^5 +...
%o (PARI) {a(n)=polcoeff(sum(m=0,n,((3*m+2)*x)^m/(1+(3*m+2)*x +x*O(x^n))^m), n)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A187735, A014479, A187738, A221160, A221161, A187740.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 03 2013
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