OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..306
FORMULA
Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5.
(2) A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5 is the e.g.f. of A251576.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251576.
E.g.f.: -LambertW(-6*x) * (6 + LambertW(-6*x))^5 / (x*6^6). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 8*x^2/2! + 156*x^3/3! + 5160*x^4/4! + 245976*x^5/5! +...
such that A(x) = exp( 6*x*A(x) * G(x*A(x))^5 ) / G(x*A(x))^5
where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(6*x*G(x)^5)/G(x)^5,
F(x) = 1 + x + 6*x^2/2! + 96*x^3/3! + 2736*x^4/4! + 115056*x^5/5! +...
is the e.g.f. of A251576.
MATHEMATICA
Table[6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)
PROG
(PARI) {a(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(G=1, A=1); for(i=1, n, G=1+x*G^6 +x*O(x^n));
for(i=1, n, A = exp(6*x*A * subst(G^5, x, x*A) ) / subst(G^5, x, x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [6^(n - 4)*(n + 1)^(n - 6)*(125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296): n in [0..50]]; // G. C. Greubel, Nov 13 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2014
STATUS
approved