OFFSET
1,5
FORMULA
G.f. A(x) satisfies:
(1) 1 - x = Sum_{n>=0} ( x^(5*n) + (-1)^n*A(x) )^n.
(2) 1 - x = Sum_{n>=0} x^(5*n^2) / (1 + (-1)^n*x^(5*n)*A(x))^(n+1).
EXAMPLE
G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + 8*x^10 + 13*x^11 + 21*x^12 + 32*x^13 + 46*x^14 + 66*x^15 + ...
where
1 - x = 1 + (x^5 - A(x)) + (x^10 + A(x))^2 + (x^15 - A(x))^3 + (x^20 + A(x))^4 + (x^25 - A(x))^5 + (x^30 + A(x))^6 + ...
Also,
1 - x = 1/(1 + A(x)) + x^5/(1 - x^5*A(x))^2 + x^20/(1 + x^10*A(x))^3 + x^45/(1 - x^15*A(x))^4 + x^80/(1 + x^20*A(x))^5 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, (x^(5*m) + (-1)^m*x*Ser(A))^m ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=0, sqrtint(#A\5), x^(5*m^2)/(1 + (-x)^(5*m)*x*Ser(A))^(m+1) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 05 2022
STATUS
approved