OFFSET
0,3
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..322
FORMULA
Let G(x) = 1 + x*G(x)^4 be the g.f. of A002293, then the e.g.f. A(x) of this sequence satisfies:
(1) A(x) = exp( 4*x*A(x) * G(x*A(x))^3 ) / G(x*A(x))^3.
(2) A(x) = F(x*A(x)) where F(x) = exp(4*x*G(x)^3)/G(x)^3 is the e.g.f. of A251574.
(3) a(n) = [x^n/n!] F(x)^(n+1)/(n+1) where F(x) is the e.g.f. of A251574.
E.g.f.: -LambertW(-4*x) * (4 + LambertW(-4*x))^3 / (256*x). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 82*x^3/3! + 1856*x^4/4! + 59904*x^5/5! + ...
such that A(x) = exp( 4*x*A(x) * G(x*A(x))^3 ) / G(x*A(x))^3
where G(x) = 1 + x*G(x)^3 is the g.f. of A002293:
G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 + ...
RELATED SERIES.
Note that A(x) = F(x*A(x)) where F(x) = exp(4*x*G(x)^3)/G(x)^3,
F(x) = 1 + x + 4*x^2/2! + 40*x^3/3! + 712*x^4/4! + 18784*x^5/5! + ...
is the e.g.f. of A251574.
MATHEMATICA
Table[4^(n - 2)*(n + 1)^(n - 4)*(3*n^2 + 13*n + 16), {n, 0, 50}] (* G. C. Greubel, Nov 13 2017 *)
PROG
(PARI) {a(n) = 4^(n-2) * (n+1)^(n-4) * (3*n^2 + 13*n + 16) }
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = local(G=1, A=1); for(i=1, n, G=1+x*G^4 +x*O(x^n)); for(i=1, n, A = exp(4*x*A * subst(G^3, x, x*A) ) / subst(G^3, x, x*A) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(Magma) [4^(n - 2)*(n + 1)^(n - 4)*(3*n^2 + 13*n + 16): n in [0..50]]; // G. C. Greubel, Nov 13 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 06 2014
STATUS
approved