A355152 G.f. A(x) satisfies: 0 = Sum_{n=-oo..+oo} x^(n^2) * (x^n - (-1)^n*A(x))^(n+1).

1, 2, 6, 21, 77, 310, 1308, 5710, 25605, 117232, 545719, 2575126, 12289906, 59218913, 287699288, 1407690851, 6930731921, 34311193558, 170691635544, 852874403933, 4278234997610, 21537155141117, 108771673373067, 550966271295019, 2798396004637028, 14248670155477872

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..400

Formula

G.f. A(x) satisfies:

(1) 0 = Sum_{n=-oo..+oo} x^(n^2) * (x^n - (-1)^n*A(x))^(n+1).

(2) 0 = Sum_{n=-oo..+oo} x^(n*(2*n-1)) / (1 - (-x)^n*A(x))^(n-1).

a(n) ~ c * d^n / n^(3/2), where d = 5.4057072074645439585749668665909363742748... and c = 0.8797898450003270547125706339934138835467... - Vaclav Kotesovec, Jun 22 2022

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 21*x^3 + 77*x^4 + 310*x^5 + 1308*x^6 + 5710*x^7 + 25605*x^8 + 117232*x^9 + 545719*x^10 + ...
where
0 = ... + x^16/(x^(-4) - A(x))^3 + x^9/(x^(-3) + A(x))^2 + x^4/(x^(-2) - A(x)) + x + (1 - A(x)) + x*(x + A(x))^2 + x^4*(x^2 - A(x))^3 + x^9*(x^3 + A(x))^4 + x^16*(x^4 - A(x))^5 + ...

Prog

(PARI) {a(n) = my(A=[1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(n+1));
A[#A] = polcoeff( sum(n=-t, t, x^(n^2) * (x^n - (-1)^n*Ser(A))^(n+1) ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A355153.

Sequence in context: A150188 A150189 A144169 * A363813 A124292 A277221

Adjacent sequences: A355149 A355150 A355151 * A355153 A355154 A355155

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2022

Status

approved