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, 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

Paul D. Hanna, Table of n, a(n) for n = 1..401

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), ", "))

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Sequence in context: A092305 A196166 A073604 * A210990 A004194 A097590

Adjacent sequences: A370029 A370030 A370031 * A370033 A370034 A370035

Keyword

sign

Author

Paul D. Hanna, Feb 11 2024

Status

approved