A355156 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n*(n+1)/2) * ((1+x)^n + 2*A(x))^(n+1).

1, 3, 7, 34, 156, 635, 2966, 14327, 66047, 314700, 1560159, 7710390, 38251395, 192549444, 974601369, 4952273462, 25349634715, 130414412847, 673028519274, 3487306501970, 18139905710007, 94640387701755, 495160150021161, 2597800048395361, 13661957722514619

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..400

Formula

G.f. A(x) satisfies:

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

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

a(n) ~ c * d^n / n^(3/2), where d = 5.5806... and c = 0.3849... - Vaclav Kotesovec, Jun 22 2022

Example

G.f.: A(x) = x + 3*x^2 + 7*x^3 + 34*x^4 + 156*x^5 + 635*x^6 + 2966*x^7 + 14327*x^8 + 66047*x^9 + 314700*x^10 + 1560159*x^11 + ...
where
2 = ... + x^6/(1/(1+x)^4 + 2*A(x))^3 - x^3/(1/(1+x)^3 + 2*A(x))^2 - x/(1/(1+x)^2 + 2*A(x)) + 1 + (1 + 2*A(x)) - x*((1+x) + 2*A(x))^2 - x^3*((1+x)^2 + 2*A(x))^3 + x^6*((1+x)^3 + 2*A(x))^4 +--+ ...

Prog

(PARI) {a(n) = my(A=[0, 1], t); for(i=1, n, A=concat(A, 0); t=ceil(sqrt(2*n+9));
A[#A] = -polcoeff( sum(n=-t, t, (-x)^(n*(n+1)/2) * ((1+x)^n + 2*Ser(A))^(n+1) ), #A-1)/2); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A355155.

Sequence in context: A024496 A081890 A192880 * A365140 A318444 A334314

Adjacent sequences: A355153 A355154 A355155 * A355157 A355158 A355159

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2022

Status

approved