OFFSET
0,2
COMMENTS
The g.f. of this sequence is motivated by the following identity:
Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;
here, p = x, q = -x*A(x), and r = -x.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..500
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n>=0} x^n / (1 - (-x)^(n+1)*A(x)).
(2) 1 = Sum_{n>=0} (-x)^n * A(x)^n / (1 + (-x)^(n+1)).
(3) 1 = Sum_{n>=0} x^(n^2 + 2*n) * A(x)^n * (1 + x^(2*n+2)*A(x)) / ((1 + (-x)^(n+1))*(1 - (-x)^(n+1)*A(x))).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 5*x^2 + 12*x^3 + 30*x^4 + 79*x^5 + 217*x^6 + 615*x^7 + 1789*x^8 + 5315*x^9 + 16054*x^10 + 49139*x^11 + 152056*x^12 + ...
where
1 = 1/(1 + x*A(x)) + x/(1 - x^2*A(x)) + x^2/(1 + x^3*A(x)) + x^3/(1 - x^4*A(x)) + x^4/(1 + x^5*A(x)) + x^5/(1 - x^6*A(x)) + x^6/(1 + x^7*A(x)) + ...
also,
1 = 1/(1 - x) - x*A(x)/(1 + x^2) + x^2*A(x)^2/(1 - x^3) - x^3*A(x)^3/(1 + x^4) + x^4*A(x)^4/(1 - x^5) - x^5*A(x)^5/(1 + x^6) + x^6*A(x)^6/(1 - x^7) + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(-1 + sum(m=0, #A, x^m / (1 - (-x)^(m+1)*Ser(A)) ), #A); ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 23 2023
STATUS
approved