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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, 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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..35.

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)

MATHEMATICA

Table[Binomial[n, Floor[n/3]] - Sum[Binomial[n, i], {i, 0, Floor[n/3] - 1}], {n, 0, 20}] (* 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

CROSSREFS

Sequence in context: A034776 A068791 A219968 * A076227 A186272 A092075

Adjacent sequences:  A126039 A126040 A126041 * A126043 A126044 A126045

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Dec 16 2006

STATUS

approved

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Last modified September 23 05:35 EDT 2019. Contains 327327 sequences. (Running on oeis4.)