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A370032
Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n + A(x))^n = 1 + 3*Sum_{n>=1} x^(n^2).
1
1, -1, 0, 3, -10, 21, -25, -23, 228, -737, 1479, -1245, -4352, 25206, -72761, 128245, -38615, -697798, 3109043, -8016819, 11763729, 6510069, -108216128, 403917707, -925174519, 1025709534, 2228869018, -16585014721, 53758505915, -107811969706, 69758146717, 478423936550, -2520835801152, 7208714823250
OFFSET
1,4
LINKS
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (x^n + A(x))^n = 1 + 3*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n - A(x))^(n-1) = 1 + 3*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*A(x))^n = 1 + 3*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n*A(x))^(n+1) = 1 + 3*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + x^n*A(x))^n = 0.
EXAMPLE
G.f.: A(x) = x - x^2 + 3*x^4 - 10*x^5 + 21*x^6 - 25*x^7 - 23*x^8 + 228*x^9 - 737*x^10 + 1479*x^11 - 1245*x^12 - 4352*x^13 + 25206*x^14 - 72761*x^15 + 128245*x^16 + ...
where
Sum_{n=-oo..+oo} (x^n + A(x))^n = 1 + 3*x + 3*x^4 + 3*x^9 + 3*x^16 + 3*x^25 + 3*x^36 + 3*x^49 + 3*x^64 + ...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0); A[#A] = -polcoeff( sum(m=-#A, #A, (x^m + Ser(A))^m ) - 1 - 3*sum(m=1, #A, x^(m^2) ), #A-1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 11 2024
STATUS
approved