A200314 -id:A200314 - cp's OEIS frontend

A200313 E.g.f. satisfies: A(x) = exp(x^3*A(x)^3/3!).

1, 1, 70, 28000, 33833800, 91842150400, 471920698849600, 4105733038511104000, 55918460253906250000000, 1124922893768186370457600000, 31962429471680921191680301600000, 1237813985055170041194334820761600000, 63474917512551971525535771981021376000000

Offset: 0

Paul D. Hanna, Nov 15 2011

nonn

			E.g.f.: A(x) = 1 + x^3/3! + 70*x^6/6! + 28000*x^9/9! + 33833800*x^12/12! + ...
where log(A(x)) = x^3*A(x)^3/3! and
A(x)^3 = 1 + 3*x^3/3! + 270*x^6/6! + 120960*x^9/9! + 155925000*x^12/12! + ...

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A025035, A034941, A200314, A200315.

List([0..10],n->(3*n+1)^(n-1)*Factorial(3*n)/(Factorial(n)*Factorial(3)^n)); # Muniru A Asiru, Jul 28 2018

[(3*n+1)^(n-1)*Factorial(3*n)/(6^n*Factorial(n)): n in [0..30]]; // G. C. Greubel, Jul 27 2018

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

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

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

a(n) = (3*n+1)^(n-1) * (3*n)!/(n!*(3!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^3/3!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(3*n)/(3*n)!
then a(n,m) = m*(3*n+m)^(n-1) * (3*n)!/(n!*(3!)^n).

A200315 E.g.f. satisfies: A(x) = exp(x^5*A(x)^5/5!).

Original entry on oeis.org

1, 1, 1386, 32288256, 4527372986136, 2373840824586206976, 3532226719132271834449776, 12455133709483299692008910094336, 91656142095228409912231665590704016256, 1280796898530759870923631204720457656538791936

Offset: 0

Paul D. Hanna, Nov 15 2011

nonn

			E.g.f.: A(x) = 1 + x^5/5! + 1386*x^10/10! + 32288256*x^15/15! +...
where log(A(x)) = x^5*A(x)^5/5! and
A(x)^5 = 1 + 5*x^5/5! + 9450*x^10/10! + 252252000*x^15/15! + 38192529375000*x^20/20! +...

Cf. A034941, A200313, A200314.

{a(n)=(5*n)!*polcoeff(1/x*serreverse(x*(exp(-x^5/5!+x*O(x^(5*n))))),5*n)}

PARI
```
{a(n)=(5*n+1)^(n-1)*(5*n)!/(n!*(5!)^n)}
```

a(n) = (5*n+1)^(n-1) * (5*n)! / (n!*(5!)^n).
E.g.f.: (1/x)*Series_Reversion( x*exp(-x^5/5!) ).
Powers of e.g.f.: define a(n,m) by A(x)^m = Sum_{n>=0} a(n,m)*x^(5*n)/(5*n)!
then a(n,m) = m*(5*n+m)^(n-1) * (5*n)!/(n!*(5!)^n).

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A200313 E.g.f. satisfies: A(x) = exp(x^3*A(x)^3/3!).

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A200315 E.g.f. satisfies: A(x) = exp(x^5*A(x)^5/5!).

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