OFFSET
0,2
COMMENTS
The g.f. A(x) 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*A(x), q = -x*A(x)^2, and r = x.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following formulas.
(1) 1 = Sum_{n>=0} x^n * A(x)^n / (1 + x^(n+1)*A(x)^2).
(2) 1 = Sum_{n>=0} (-1)^n * x^n * A(x)^(2*n) / (1 - x^(n+1)*A(x)).
(3) x = Sum_{n>=1} (-1)^(n-1) * x^(n^2) * A(x)^(3*(n-1)) * (1 + x^(2*n)*A(x)^3) / ((1 - x^n*A(x))*(1 + x^n*A(x)^2)).
EXAMPLE
G.f.: A(x) = 1 + 2*x + 7*x^2 + 32*x^3 + 165*x^4 + 915*x^5 + 5321*x^6 + 32013*x^7 + 197589*x^8 + 1244127*x^9 + 7960163*x^10 + ...
where
1 = 1/(1 + x*A(x)^2) + x*A(x)/(1 + x^2*A(x)^2) + x^2*A(x)^2/(1 + x^3*A(x)^2) + x^3*A(x)^3/(1 + x^4*A(x)^2) + x^4*A(x)^4/(1 + x^5*A(x)^2) + ...
also,
1 = 1/(1 - x*A(x)) - x*A(x)^2/(1 - x^2*A(x)) + x^2*A(x)^4/(1 - x^3*A(x)) - x^3*A(x)^6/(1 - x^4*A(x)) + x^4*A(x)^8/(1 - x^5*A(x)) -+ ...
PROG
(PARI) {a(n, k=2) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-1 + sum(n=0, #A, x^n * Ser(A)^((k-1)*n) / (1 + x^(n+1)*Ser(A)^k ) ), #A)); A[n+1]}
for(n=0, 30, print1(a(n, 2), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 19 2023
STATUS
approved