OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^(5*n) - A(x))^n.
(2) 1 = Sum_{n>=0} (1+x)^(5*n^2) / (1 + (1+x)^(5*n)*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 5*x + 35*x^2 + 1110*x^3 + 61830*x^4 + 4607001*x^5 + 422112085*x^6 + 45521033720*x^7 + 5625206604320*x^8 + 782244114339935*x^9 + ...
such that
1 = 1 + ((1+x)^5 - A(x)) + ((1+x)^10 - A(x))^2 + ((1+x)^15 - A(x))^3 + ((1+x)^20 - A(x))^4 + ((1+x)^25 - A(x))^5 + ((1+x)^30 - A(x))^6 + ((1+x)^35 - A(x))^7 + ...
Also,
1 = 1/(1 + A(x)) + (1+x)^5/(1 + (1+x)^5*A(x))^2 + (1+x)^20/(1 + (1+x)^10*A(x))^3 + (1+x)^45/(1 + (1+x)^15*A(x))^4 + (1+x)^80/(1 + (1+x)^20*A(x))^5 + (1+x)^125/(1 + (1+x)^25*A(x))^6 + ...
RELATED SERIES.
The logarithmic derivative of the g.f. begins
A'(x)/A(x) = 5 + 45*x + 2930*x^2 + 225545*x^3 + 21445630*x^4 + 2388480630*x^5 + 303204843520*x^6 + 43104182972905*x^7 + 6777636393880895*x^8 + ...
the coefficients of which are all divisible by 5:
(1/5) * A'(x)/A(x) = 1 + 9*x + 586*x^2 + 45109*x^3 + 4289126*x^4 + 477696126*x^5 + 60640968704*x^6 + 8620836594581*x^7 + 1355527278776179*x^8 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ((1+x)^(5*m) - Ser(A))^m ) )[#A] ); H=A; A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 14 2018
STATUS
approved