A085363 a(0)=1, for n>0: a(n) = 4*9^(n-1) - (1/2)*Sum_{i=1..n-1} a(i)*a(n-i).

1, 4, 28, 212, 1676, 13604, 112380, 940020, 7936620, 67494980, 577309148, 4961187092, 42801458764, 370478720356, 3215827927228, 27982214082612, 244004165618220, 2131710838837380, 18654504783815580, 163488269572628820

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Jun 25 2003

Keywords

• easy
• nonn
• changed

Comments

Apparently, the number of 2-D directed walks of semilength n starting at (0,0) and ending on the X-axis using steps NE, SE, NW and SW avoiding adjacent NW/SE and adjacent NE/SW. - David Scambler, Jun 20 2013

Form an array with m(0,n) = m(n,0) = 2^n; m(i,j) equals the sum of the terms to the left of m(i,j) and the sum of the terms above m(i,j), which is m(i,j) = Sum_{k-0..j-1} m(i,k) + Sum_{k=0..i-1} m(k,j). m(n,n) = a(n). - J. M. Bergot, Jul 10 2013

From G. C. Greubel, May 22 2020: (Start)

This sequence is part of a class of sequences, for m >= 0, with the properties:

a(n) = 2*m*(4*m+1)^(n-1) - (1/2)*Sum_{k=1..n-1} a(k)*a(n-k).

a(n) = Sum_{k=0..n} m^k * binomial(n-1, n-k) * binomial(2*k, k).

a(n) = (2*m) * Hypergeometric2F1(-n+1, 3/2; 2; -4*m), for n > 0.

n*a(n) = 2*((2*m+1)*n - (m+1))*a(n-1) - (4*m+1)*(n-2)*a(n-2).

(4*m + 1)^n = Sum_{k=0..n} Sum_{j=0..k} a(j)*a(k-j).

G.f.: sqrt( (1 - t)/(1 - (4*m+1)*t) ).

This sequence is the case of m=2. (End)

The number of elements in the free group on two generators of length 2n that are zero exponent sum. - Tey Berendschot, Aug 09 2021

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Kees A.S. Immink and Kui Cai, Properties and constructions of constrained codes for DNA-based data storage, IEEE Access, vol. 8, no. 1, pp. 49523-49531, 2020, page 49529.
• John Machacek, Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats, preprint arXiv:2105.02417 [math.CO], 2021.

Crossrefs

Cf. A001019 (9^n), A084771, A085362, A085364, diagonal of A348595.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( Sqrt((1-x)/(1-9*x)) )); // G. C. Greubel, May 23 2020

Maple

seq(coeff(series(sqrt((1-x)/(1-9*x)), x, n+1), x, n), n = 0..30); # G. C. Greubel, May 23 2020

Mathematica

CoefficientList[Series[Sqrt[(1-x)/(1-9x)], {x, 0, 25}], x]

PARI

my(x='x+O('x^66)); Vec(sqrt((1-x)/(1-9*x)) ) \\ Joerg Arndt, May 10 2013

Sage

def A085363_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt((1-x)/(1-9*x)) ).list()
A085363_list(30) # G. C. Greubel, May 23 2020

Formula

G.f.: sqrt((1-x)/(1-9*x)).

Sum_{i=0..n} Sum_{j=0..i} a(j)*a(i-j) = 9^n.

From Vladeta Jovovic, Oct 10 2003: (Start)

First differences of A084771.

a(n) = Sum_{k=1..n} 2^k * binomial(n-1, k-1) * binomial(2*k, k). (End)

D-finite with recurrence n*a(n) = (10*n-6)*a(n-1) - (9*n-18)*a(n-2). - Vladeta Jovovic, Jul 16 2004

a(n) ~ 2*sqrt(2)*3^(2*n-1)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012

a(0) = 1; a(n) = (4/n) * Sum_{k=0..n-1} (n+k) * a(k). - Seiichi Manyama, Mar 28 2023

From Seiichi Manyama, Aug 22 2025: (Start)

a(n) = (1/4)^n * Sum_{k=0..n} 9^k * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).

a(n) = Sum_{k=0..n} (-2)^k * 9^(n-k) * binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)