OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..293
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) * G(x*A(x))^8 ) / G(x*A(x))^8.
(2) A(x) = F(x*A(x)) where F(x) = exp(9*x*G(x)^8)/G(x)^8 is the e.g.f. of A251579.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251579.
E.g.f.: -LambertW(-9*x) * (9 + LambertW(-9*x))^8 / (x*9^9). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 11*x^2/2! + 312*x^3/3! + 15261*x^4/4! + 1082784*x^5/5! +...
such that A(x) = exp( 9*x*A(x) * G(x*A(x))^8 ) / G(x*A(x))^8
where G(x) = 1 + x*G(x)^9 is the g.f. of A062994:
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(9*x*G(x)^8)/G(x)^8,
F(x) = 1 + x + 9*x^2/2! + 225*x^3/3! + 10017*x^4/4! + 656289*x^5/5! +...
is the e.g.f. of A251579.
MATHEMATICA
Table[9^(n - 7)*(n + 1)^(n - 9)*(262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)
PROG
(PARI) {a(n) = 9^(n-7) * (n+1)^(n-9) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(G=1, A=1); for(i=1, n, G=1+x*G^9 +x*O(x^n));
for(i=1, n, A = exp(9*x*A * subst(G^8, x, x*A) ) / subst(G^8, x, x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [9^(n - 7)*(n + 1)^(n - 9)*(262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969): n in [0..10]]; // G. C. Greubel, Nov 13 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2014
STATUS
approved