A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

1, 0, 0, 1, 1, 1, 3, 5, 7, 14, 26, 43, 79, 148, 264, 483, 903, 1664, 3080, 5771, 10795, 20209, 38059, 71799, 135569, 256762, 487310, 925981, 1762841, 3361897, 6419595, 12275301, 23505143, 45061424, 86485016, 166176499, 319630115, 615387675, 1185940209, 2287527119, 4416083429

Offset

0,7

Links

Table of n, a(n) for n=0..40.

Formula

a(0) = 1, a(1) = a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

a(n) ~ 2^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 30 2021

From Seiichi Manyama, Sep 26 2024: (Start)

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

a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-2*k-1,n-3*k) / (k+1). (End)

Mathematica

nmax = 40; A[_] = 0; Do[A[x_] = 1 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 40}]

Crossrefs

Cf. A002212, A023426, A023431, A090345, A216604 (partial sums), A346504, A366588, A376490.

Sequence in context: A308591 A309832 A319784 * A061390 A227685 A051445

Adjacent sequences: A346500 A346501 A346502 * A346504 A346505 A346506

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 21 2021

Status

approved