OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..330
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) * G(x*A(x))^2 ) / G(x*A(x))^2.
(2) A(x) = F(x*A(x)) where F(x) = exp(3*x*G(x)^2)/G(x)^2 is the e.g.f. of A251573.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251573.
E.g.f.: -LambertW(-3*x) * (3 + LambertW(-3*x))^2 / (27*x). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 54*x^3/3! + 945*x^4/4! + 23328*x^5/5! +...
such that A(x) = exp( 3*x*A(x) * G(x*A(x))^2 ) / G(x*A(x))^2
where G(x) = 1 + x*G(x)^3 is the g.f. of A001764:
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(3*x*G(x)^2)/G(x)^2,
F(x) = 1 + x + 3*x^2/2! + 21*x^3/3! + 261*x^4/4! + 4833*x^5/5! +...
is the e.g.f. of A251573.
MATHEMATICA
Table[3^(n - 1)*(n + 1)^(n - 3)*(n + 3), {n, 0, 20}] (* G. C. Greubel, Nov 12 2017 *)
PROG
(PARI) {a(n) = 3^(n-1) * (n+1)^(n-3) * (n+3)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(G=1, A=1); for(i=1, n, G=1+x*G^3 +x*O(x^n));
for(i=1, n, A = exp(3*x*A * subst(G^2, x, x*A) ) / subst(G^2, x, x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2014
STATUS
approved