OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..235
FORMULA
Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8).
(2) A(x) = F(x*A(x)^8) where F(x) = exp(9*x*G(x)^8)/G(x) is the e.g.f. of A251669.
(3) A(x) = ( Series_Reversion( x*G(x)^8 / exp(72*x*G(x)^8) )/x )^(1/8).
E.g.f.: (-LambertW(-72*x)/(72*x))^(1/8) * (1 + LambertW(-72*x)/72). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + 8*x + 1215*x^2/2! + 400950*x^3/3! + 207202941*x^4/4! + 146509780644*x^5/5! +...
such that A(x) = exp( 9*x*A(x)^8 * G(x*A(x)^8)^8 ) / G(x*A(x)^8),
where G(x) = 1 + x*G(x)^9 is the g.f. A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
Also, e.g.f. A(x) satisfies A(x) = F(x*A(x)^8) where
F(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...
F(x) = exp( 9*x*G(x)^8 ) / G(x) is the e.g.f. of A251669.
MATHEMATICA
Table[(7*n + 1)*(8*n + 1)^(n - 2)*9^n, {n, 0, 50}] (* G. C. Greubel, Nov 14 2017 *)
PROG
(PARI) {a(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=local(G=1, A=1); for(i=0, n, G = 1 + x*G^9 +x*O(x^n));
A = ( serreverse( x*G^8 / exp(72*x*G^8) )/x )^(1/8); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [(7*n + 1)*(8*n + 1)^(n - 2)*9^n: n in [0..50]]; // G. C. Greubel, Nov 14 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 07 2014
STATUS
approved