A355155 G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n*(n+1)/2) * ((1+x)^(n+1) - 2*A(x))^n.

1, 1, 7, 28, 184, 1024, 6676, 42367, 282765, 1897203, 13004369, 89991470, 630242521, 4450613382, 31683411117, 227041605009, 1636747514265, 11860982110191, 86356914006201, 631382010617369, 4633749074928932, 34124201919637479, 252086975581304199

Offset

1,3

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+1) - 2*A(x))^n.

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

a(n) ~ c * d^n / n^(3/2), where d = 7.89889... and c = 0.06269... - Vaclav Kotesovec, Jun 22 2022

Example

G.f.: A(x) = x + x^2 + 7*x^3 + 28*x^4 + 184*x^5 + 1024*x^6 + 6676*x^7 + 42367*x^8 + 282765*x^9 + 1897203*x^10 + 13004369*x^11 + ...
where
2 = ... + x^6/(1/(1+x)^3 - 2*A(x))^4 - x^3/(1/(1+x)^2 - 2*A(x))^3 - x/(1/(1+x) - 2*A(x))^2 + 1/(1 - 2*A(x)) + 1 - x*((1+x)^2 - 2*A(x)) - x^3*((1+x)^3 - 2*A(x))^2 + x^6*((1+x)^4 - 2*A(x))^3 +--+ ...

Prog

(PARI) {a(n) = my(A=[0, 1], t); for(i=1, n, A=concat(A, 0); t = ceil(sqrt(2*n+4));
A[#A] = -polcoeff( sum(n=-t, t, (-x)^(n*(n+1)/2) * ((1+x)^(n+1) - 2*Ser(A))^n ), #A-1)/2); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A355156.

Sequence in context: A224663 A378554 A203296 * A058822 A054369 A185360

Adjacent sequences: A355152 A355153 A355154 * A355156 A355157 A355158

Keyword

nonn

Author

Paul D. Hanna, Jun 21 2022

Status

approved