%I #13 Sep 23 2013 09:54:33

%S 1,2,16,116,1456,18272,315424,5592512,123304192,2814746624,

%T 75639399424,2108241486848,66872341633024,2198914617257984,

%U 80437062279012352,3046047243283570688,126259635313097506816,5408763597941368291328,250569314672586154835968

%N E.g.f.: A(x) = 1/(1 - 6*x^2)^(1 + 1/(3*x)).

%C More generally, we have the identity:

%C Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here x=2*x, y=3*x.

%H Vincenzo Librandi, <a href="/A193289/b193289.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: A(x) = Sum_{n>=0} 2^n*x^n/n! * Product_{k=1..n} (1 + 3*k*x).

%F a(n) ~ n! * 6^(n/2)*(n/2)^sqrt(2/3)/(2*Gamma(1+sqrt(2/3))). - _Vaclav Kotesovec_, Sep 22 2013

%e E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 116*x^3/3! + 1456*x^4/4! + 18272*x^5/5! +...

%e where A(x) satisfies:

%e A(x)^(3*x/(1+3*x)) = 1 + 6*x^2 + 36*x^4 + 216*x^6 +...+ 6^n*x^(2*n) +...

%e Also,

%e A(x) = 1 + 2*x*(1+3*x) + 4*x^2*(1+3*x)*(1+6*x)/2! + 8*x^3*(1+3*x)*(1+6*x)*(1+9*x)/3! + 16*x^4*(1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)/4! +...

%e The logarithm begins:

%e log(A(x)) = 2*x + 6*x^2 + 2*6*x^3/2 + 6^2*x^4/2 + 2*6^2*x^5/3 + 6^3*x^6/3 + 2*6^3*x^7/4 + 6^4*x^8/4 + 2*6^4*x^9/5 + 6^5*x^10/5 +...

%t CoefficientList[Series[1/(1-6*x^2)^(1+1/(3*x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Sep 22 2013 *)

%o (PARI) {a(n)=n!*polcoeff(1/(1 - 6*x^2 +x^2*O(x^n))^((1+3*x)/(3*x)),n)}

%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n,2^m*x^m/m!*prod(k=1,m,1+3*k*x+x*O(x^n))),n)}

%Y Cf. A193281, A193287, A193288, A193290.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 07 2011