A307411 G.f. A(x) satisfies: A(x) = 1 + x*A(x)*(1 + 2*x*A(x))/(1 - x*A(x) - x^2*A(x)^2).

1, 1, 4, 14, 60, 267, 1254, 6071, 30156, 152714, 785682, 4094752, 21573258, 114709363, 614777462, 3317589966, 18011350796, 98307220409, 539121535194, 2969177051678, 16415395615190, 91070109305056, 506843759000184, 2828968117483929, 15831944500607010, 88818114923080102

Offset: 0

Author

Ilya Gutkovskiy, Apr 07 2019

Keywords

• nonn

Crossrefs

Cf. A000204, A049124, A307412, A307413.

Programs

Mathematica

terms = 25; A[] = 0; Do[A[x] = 1 + x A[x] (1 + 2 x A[x])/(1 - 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[LucasL[k] 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 - x - x^2)/(1 + x^2), {x, 0, terms}], x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} Lucas(k)*x^k*A(x)^k, where Lucas = A000204.

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