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A190666 Number of walks from (0,0) to (n+3,n) which take steps from {E, N, NE}. 4
1, 9, 61, 377, 2241, 13073, 75517, 433905, 2485825, 14218905, 81270333, 464387817, 2653649025, 15167050785, 86716873725, 495998874593, 2838240338817, 16248650965289, 93065296937533, 533285164334169, 3057236753252161, 17534423944871729, 100609937775369981 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

S. Gao, H. Niederhausen, Counting New Lattice Paths and Walks with Several Step Vectors (submitted to Congr. Numer.). - Shanzhen Gao, May 25 2011

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..400

Shanzhen Gao, Keh-Hsun Chen, Counting Lattice Paths and Walks with Several Step Vectors, FCS 2014

FORMULA

a(n) = Sum_{k=0..n} C(n,k) * C(n+k+3,k+3) = A113139 (n+3,3). - Alois P. Heinz, Jun 01 2011

G.f.: (-1+3*x-x^2+(1-6*x+6*x^2-x^3)/sqrt(x^2-6*x+1))/(2*x^3). - Alois P. Heinz, Jun 03 2011

Recurrence: n*(n+3)*a(n) = (5*n^2 + 15*n + 16)*a(n-1) + (5*n^2 - 5*n+6)*a(n-2) - (n-2)*(n+1)*a(n-3). - Vaclav Kotesovec, Oct 20 2012

a(n) ~ sqrt(1632+1154*sqrt(2))*(3+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012

From Peter Bala, Mar 02 2017: (Start)

a(n) = 1/2^(n+1)*Sum_{k = 3..inf} 1/2^k*binomial(n + k, k)*binomial(n + k, n + 3).

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

n*(n + 3)*(2*n + 1)*a(n) = 6*(n + 1)*(2*n^2 + 4*n + 3)*a(n-1) - (n - 1)*(n + 2)*(2*n + 3)*a(n-2) with a(0) = 1 and a(1) = 9. (End)

a(n) = (-1)^n*hypergeom([-n, n+4], [1], 2). - Peter Luschny, Mar 02 2017

MAPLE

b:= proc(i, j) option remember;

      if i<0 or j<0 then 0

    elif i=0 and j=0 then 1

    else b(i-1, j) +b(i, j-1) +b(i-1, j-1)

      fi

    end:

a:= n-> b(n+3, n):

seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2011

MATHEMATICA

b[i_, j_] /; i<0 || j<0 = 0; b[0, 0] = 1; b[i_, j_]:= b[i, j]= b[i-1, j] + b[i, j-1] + b[i-1, j-1]; a[n_] := b[n+3, n]; Table[a[n], {n, 0, 20}] (* Jean-Fran├žois Alcover, Jun 01 2011, after Maple prog. *)

CoefficientList[Series[(-1+3*x-x^2+(1-6*x+6*x^2-x^3)/Sqrt[x^2-6*x+1])/(2*x^3), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

Table[(-1)^n Hypergeometric2F1[-n, n+4, 1, 2], {n, 0, 22}] (* Peter Luschny, Mar 02 2017 *)

CROSSREFS

Cf. A113139, A002002, A026002.

Sequence in context: A305783 A005060 A125346 * A016200 A001454 A243877

Adjacent sequences:  A190663 A190664 A190665 * A190667 A190668 A190669

KEYWORD

nonn,walk,easy

AUTHOR

Shanzhen Gao, May 25 2011

STATUS

approved

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Last modified August 25 13:50 EDT 2019. Contains 326324 sequences. (Running on oeis4.)