A307412 G.f. A(x) satisfies: A(x) = 1 + x*A(x)/(1 - 2*x*A(x) - x^2*A(x)^2).

1, 1, 3, 12, 53, 250, 1234, 6295, 32925, 175616, 951596, 5223658, 28987546, 162349759, 916502869, 5209630108, 29792226533, 171284524184, 989460348216, 5740230703588, 33429379234924, 195361236443008, 1145312096390408, 6733896357727242, 39697441350016170, 234596104853541967

Offset

0,3

Links

Table of n, a(n) for n=0..25.

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Pell(k)*x^k*A(x)^k, where Pell = A000129.

G.f.: A(x) = (1/x)*Series_Reversion(x*(1 - 2*x - x^2)/(1 - x - x^2)).

a(n) ~ sqrt((1 + 2^(1/3))*(4 + 7*2^(1/3))) * (2 + 3/2^(2/3) + 3/2^(1/3))^n / (3 * sqrt(Pi) * (2*n)^(3/2)). - Vaclav Kotesovec, Nov 05 2021

Mathematica

terms = 25; A[_] = 0; Do[A[x_] = 1 + x A[x]/(1 - 2 x A[x] - x^2 A[x]^2) + O[x]^(terms + 1) // Normal, {terms + 1}]; CoefficientList[A[x], x]
terms = 26; A[_] = 0; Do[A[x_] = 1 + Sum[Fibonacci[k, 2] x^k A[x]^k, {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 26; CoefficientList[1/x InverseSeries[Series[x (1 - 2 x - x^2)/(1 - x - x^2), {x, 0, terms}], x], x]

Crossrefs

Cf. A000129, A049124, A307411, A307413.

Sequence in context: A262442 A026781 A110122 * A302188 A060460 A306525

Adjacent sequences: A307409 A307410 A307411 * A307413 A307414 A307415

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 07 2019

Status

approved