OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..250
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 13*x^5 + 61*x^6 + 133*x^7 + 449*x^8 + 1825*x^9 + 11497*x^10 + 29905*x^11 + 121529*x^12 +...
Define F(n,x) with F(0,x) = A(x), where for n>0 we have
F(n-1,x) = 1/(1 - (n*x*F(n,x))^n )^(1/n), or equivalently
F(n,x) = (1 - 1/F(n-1,x)^n)^(1/n) / (n*x),
then
F(1,x) = 1 + 2*x^2 + 3*2*x^4 + 6^2*x^5 + 5*2^2*x^6 + 3^3*2^3*x^7 +...
F(2,x) = 1 + 3^2*x^3 + 6*3^3*x^6 + 12^3*x^7 + 14*3^5*x^9 + 4^4*3^5*x^10 +...
F(3,x) = 1 + 4^3*x^4 + 10*4^5*x^8 + 20^4*x^9 + 30*4^8*x^12 + 5^5*4^7*x^13 +...
F(4,x) = 1 + 5^4*x^5 + 15*5^7*x^10 + 30^5*x^11 + 55*5^11*x^15 + 6^6*5^9*x^16 +...
F(5,x) = 1 + 6^5*x^6 + 21*6^9*x^12 + 42^6*x^13 + 91*6^14*x^18 + 7^7*6^11*x^19 +...
....
F(1,x)^2 = 1 + 2^2*x^2 + 2^4*x^4 + 2^3*3^2*x^5 + 2^6*x^6 + 2*3^2*2^5*x^7 +...
F(2,x)^3 = 1 + 3^3*x^3 + 3^6*x^6 + 3^4*4^3*x^7 + 3^9*x^9 + 2*4^3*3^7*x^10 +...
F(3,x)^4 = 1 + 4^4*x^4 + 4^8*x^8 + 4^5*5^4*x^9 + 4^12*x^12 + 2*5^4*4^9*x^13 +...
F(4,x)^5 = 1 + 5^5*x^5 + 5^10*x^10 + 5^6*6^5*x^11 + 5^15*x^15 + 2*6^5*5^11*x^16 +...
F(5,x)^6 = 1 + 6^6*x^6 + 6^12*x^12 + 6^7*7^6*x^13 + 6^18*x^18 + 2*7^6*6^13*x^19 +...
...
PROG
(PARI) {a(n)=local(A=1+n*x); for(k=0, n-1, A=(1 - ((n-k)*x*A)^(n-k) +x*O(x^n))^(-1/(n-k))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 01 2014
STATUS
approved