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%I #26 Mar 01 2024 02:05:10
%S 1,2,21,270,4212,77760,1662120,40415760,1102248000,33331979520,
%T 1107097891200,40069801094400,1569793384051200,66185883219456000,
%U 2988292627358438400,143855017177487616000,7355369573944584192000,398090614491857903616000,22737098558477268725760000
%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+1) * 3^(n-1) * n!/2 for n>0 with a(0)=1.
%F E.g.f.: (2 - 8*x + 15*x^2)/(2*(1-3*x)^2).
%e O.g.f.: A(x) = 1 + 2*x + 21*x^2 + 270*x^3 + 4212*x^4 + 77760*x^5 +...
%e where
%e A(x) = 1 + 2*x/(1+2*x) + 5^2*x^2/(1+5*x)^2 + 8^3*x^3/(1+8*x)^3 + 11^4*x^4/(1+11*x)^4 + 14^5*x^5/(1+14*x)^5 +...
%e E.g.f.: E(x) = 1 + 2*x + 21*x^2/2! + 270*x^3/3! + 4212*x^4/4! + 77760*x^5/5! +...
%e where
%e E(x) = 1 + 2*x + 21/2*x^2 + 45*x^3 + 351/2*x^4 + 648*x^5 + 4617/2*x^6 +...
%e which is the expansion of: (2 - 8*x + 15*x^2) / (2 - 12*x + 18*x^2).
%t Join[{1},Table[(3n+1)3^(n-1) n!/2,{n,20}]] (* _Harvey P. Dale_, Feb 10 2015 *)
%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), ", "))
%o (PARI) {a(n) = if(n==0,1,(3*n+1)*3^(n-1)*n!/2)}
%o for(n=0, 20, print1(a(n), ", "))
%Y Cf. A229039, A187735, A187738, A187739, A221160, A221161, A187740.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 11 2013
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