A200313 - OEIS

%I #25 Sep 08 2022 08:46:00

%S 1,1,70,28000,33833800,91842150400,471920698849600,

%T 4105733038511104000,55918460253906250000000,

%U 1124922893768186370457600000,31962429471680921191680301600000,1237813985055170041194334820761600000,63474917512551971525535771981021376000000

%N E.g.f. satisfies: A(x) = exp(x^3*A(x)^3/3!).

%H G. C. Greubel, <a href="/A200313/b200313.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = (3*n+1)^(n-1) * (3*n)!/(n!*(3!)^n).

%F E.g.f.: (1/x)*Series_Reversion( x*exp(-x^3/3!) ).

%F Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(3*n)/(3*n)!

%F then a(n,m) = m*(3*n+m)^(n-1) * (3*n)!/(n!*(3!)^n).

%e E.g.f.: A(x) = 1 + x^3/3! + 70*x^6/6! + 28000*x^9/9! + 33833800*x^12/12! + ...

%e where log(A(x)) = x^3*A(x)^3/3! and

%e A(x)^3 = 1 + 3*x^3/3! + 270*x^6/6! + 120960*x^9/9! + 155925000*x^12/12! + ...

%t Table[(3*n + 1)^(n - 1)*(3*n)!/(n!*(3!)^n), {n, 0, 30}] (* _G. C. Greubel_, Jul 27 2018 *)

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

%o (PARI) {a(n)=(3*n+1)^(n-1)*(3*n)!/(n!*(3!)^n)}

%o (Magma) [(3*n+1)^(n-1)*Factorial(3*n)/(6^n*Factorial(n)): n in [0..30]]; // _G. C. Greubel_, Jul 27 2018

%o (GAP) List([0..10],n->(3*n+1)^(n-1)*Factorial(3*n)/(Factorial(n)*Factorial(3)^n)); # _Muniru A Asiru_, Jul 28 2018

%Y Cf. A025035, A034941, A200314, A200315.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 15 2011