OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..100
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 1 = Sum_{n=-oo..+oo} x^n * (1 + x^(n-1))^n * (-2*A(x))^(n*(n-1)/2).
(2) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 + x^(n+1))^n * (-2*A(x))^(n*(n+1)/2).
EXAMPLE
G.f.: A(x) = 1 - x - 4*x^2 + 44*x^3 + 316*x^4 - 22695*x^5 - 769536*x^6 + 156937802*x^7 + 30299780744*x^8 - 18827264809946*x^9 - 17187430890378027*x^10 ++-- ...
such that
B(x) = Sum_{n>=1} x^n * (1 + x^(n-1))^n * (-2*A(x))^(n*(n-1)/2)
and
B(x) = -Sum_{n>=1} x^(n^2) / (1 + x^(n+1))^n * (-2*A(x))^(n*(n+1)/2),
where
B(x) = 2*x - 2*x^2 - 10*x^3 + 98*x^4 + 618*x^5 - 45552*x^6 - 1538490*x^7 + 313926892*x^8 + 60600533658*x^9 - 37654860921240*x^10 - 34374918573912040*x^11 ++-- ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(sum(m=-#A, #A, (x + x^m)^m * (-2*Ser(A))^(m*(m-1)/2) ), #A)/2); H=A; A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 07 2022
STATUS
approved