A126042 Expansion of f(x^3)/(1-x*f(x^3)), where f(x) is the g.f. of A001764, whose n-th term is binomial(3n,n)/(2n+1).

1, 1, 1, 2, 3, 4, 8, 13, 19, 38, 64, 98, 196, 337, 531, 1062, 1851, 2974, 5948, 10468, 17060, 34120, 60488, 99658, 199316, 355369, 590563, 1181126, 2115577, 3540464, 7080928, 12731141, 21430267, 42860534, 77306428, 130771376, 261542752, 473018396, 803538100

Offset

0,4

Comments

Row sums of number triangle A111373.

Interleaves T(3n,2n), T(3n+1,2n+1) and T(3n+2,2n+2) for T(n,k) = A047089(n,k).

One step forward and two steps back: number of nonnegative walks of n steps where the steps are size 1 forwards and size 2 backwards. - David Scambler, Mar 15 2011

Brown's criterion ensures that the sequence is complete (see formulae). - Vladimir M. Zarubin, Aug 05 2019

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Eric Weisstein's World of Mathematics, Brown's Criterion

Formula

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

G.f.: (1/x)*Series_Reversion( x*(1+x)^2/((1+x)^3+x^3) ). - Paul D. Hanna, Mar 15 2011

From Vladimir M. Zarubin, Aug 05 2019: (Start)

a(0) = 1, a(1) = 1, a(2) = 1 and for k>0

a(3*k) = 2*a(3*k-1),

a(3*k+1) = 2*a(3*k) - binomial(3*k,k)/(2*k+1),

a(3*k+2) = 2*a(3*k+1) - binomial(3*k+1,k)/(k+1),

where binomial(3*k,k)/(2*k+1) = A001764(k)

and binomial(3*k+1,k)/(k+1) = A006013(k). (End)

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/3)} (n-3*k+1) * binomial(n+1,k). - Seiichi Manyama, Jan 27 2024

Maple

a:= proc(n) option remember; `if`(n<4, [1$3, 2][n+1], (a(n-1)*
       2*(20*n^4-14*n^3-31*n^2-n+8)-6*(3*n-1)*(5*n-6)*a(n-2)
      +9*(n-2)*(15*n^3-48*n^2+15*n+14)*a(n-3)-54*(n-2)*(n-3)*
      (5*n^2-n-2)*a(n-4))/(2*(2*n+1)*(n+1)*(5*n^2-11*n+4)))
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Sep 07 2022

Mathematica

Table[Binomial[n, Floor[n/3]] -Sum[Binomial[n, i], {i, 0, Floor[n/3] -1}], {n, 0, 40}] (* David Callan, Oct 26 2017 *)

Prog

(PARI) {a(n)=polcoeff((1/x)*serreverse(x*(1+x)^2/((1+x)^3+x^3+x*O(x^n))), n)}
(PARI)
n=30;
{a0=1; a1=1; a2=1; for(k=1, n/3, print1(a0, ", ", a1, ", ", a2, ", ");
a0=2*a2; a1=2*a0-binomial(3*k, k)/(2*k+1); a2=2*a1-binomial(3*k+1, k)/(k+1))
} // Vladimir M. Zarubin, Aug 05 2019
(Magma) [n lt 3 select 1 else Binomial(n, Floor(n/3)) - (&+[Binomial(n, j): j in [0..Floor(n/3)-1]]): n in [0..40]]; // G. C. Greubel, Jul 30 2022
(SageMath) [binomial(n, (n//3)) - sum(binomial(n, j) for j in (0..(n//3)-1)) for n in (0..40)] # G. C. Greubel, Jul 30 2022

Crossrefs

Cf. A047089, A111373.

Sequence in context: A034776 A068791 A219968 * A076227 A186272 A361722

Adjacent sequences: A126039 A126040 A126041 * A126043 A126044 A126045

Keyword

easy,nonn

Author

Paul Barry, Dec 16 2006

Status

approved