OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..305
FORMULA
Let G(x) = 1 + x*G(x)^3 be the g.f. of A001764, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 3*x*A(x)^2 * G(x*A(x)^2)^2 ) / G(x*A(x)^2).
(2) A(x) = F(x*A(x)^2) where F(x) = exp(3*x*G(x)^2)/G(x) is the e.g.f. of A251663.
(3) A(x) = sqrt( Series_Reversion( x*G(x)^2 / exp(6*x*G(x)^2) )/x ).
E.g.f.: sqrt(-LambertW(-6*x)/(6*x))*(1+LambertW(-6*x)/6). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 27*x^2/2! + 756*x^3/3! + 32805*x^4/4! +...
such that A(x) = exp( 3*x*A(x)^2 * G(x*A(x)^2)^2 ) / G(x*A(x)^2),
where G(x) = 1 + x*G(x)^3 is the g.f. A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^2) where
F(x) = 1 + 2*x + 11*x^2/2! + 120*x^3/3! + 2061*x^4/4! + 48918*x^5/5! +...
F(x) = exp( 3*x*G(x)^2 ) / G(x) is the e.g.f. of A251663.
MATHEMATICA
Table[(n + 1)*(2*n + 1)^(n - 2)*3^n, {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)
PROG
(PARI) {a(n) = (n+1) * (2*n+1)^(n-2) * 3^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^3 +x*O(x^n));
A=( serreverse( x*G^2 / exp(6*x*G^2) )/x )^(1/2); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [(n + 1)*(2*n + 1)^(n - 2)*3^n: n in [0..50]]; // G. C. Greubel, Nov 13 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 07 2014
STATUS
approved