A393181 G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 * A(x)^2 / (1 - x).

1, 1, 1, 1, 3, 6, 10, 19, 39, 78, 154, 311, 639, 1315, 2713, 5639, 11793, 24750, 52118, 110169, 233673, 497025, 1059939, 2266089, 4855991, 10427740, 22436304, 48362711, 104427395, 225845804, 489170840, 1061015979, 2304411891, 5011201284, 10910310700

Offset

0,5

Links

Table of n, a(n) for n=0..34.

Formula

G.f.: (1 - x - sqrt(1 - x * (2 - x + 4*x^2 - 4*x^5))) / (2*x^3).

a(0) = a(1) = a(2) = 1; a(n) = Sum_{j=0..n-3} Sum_{i=0..j} a(i) * a(j-i).

Mathematica

nmax = 34; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
nmax = 34; CoefficientList[Series[(1 - x - Sqrt[1 - x (2 - x + 4 x^2 - 4 x^5)])/(2 x^3), {x, 0, nmax}], x]
a[0] = a[1] = a[2] = 1; a[n_] := a[n] = Sum[Sum[a[i] a[j - i], {i, 0, j}], {j, 0, n - 3}]; Table[a[n], {n, 0, 34}]

Crossrefs

Cf. A002212, A023431, A025243, A307970, A346503, A346504, A393182.

Sequence in context: A065614 A183324 A058356 * A270048 A295719 A178850

Adjacent sequences: A393178 A393179 A393180 * A393182 A393183 A393184

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 04 2026

Status

approved