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A092765 Consider the 1-D random walk with jumps to next-nearest neighbors. Sequence gives number of paths of length n ending at origin. 7
1, 0, 4, 6, 36, 100, 430, 1470, 5796, 21336, 82404, 312180, 1203246, 4617756, 17846686, 68974906, 267498660, 1038555024, 4040525320, 15739195680, 61399048036, 239788778760, 937536139764, 3669179504364, 14373144873774, 56350223472600, 221094286028100 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
In Lakatos-Lindenberg and Shuler besides some physical background there is an exact algebraic expression for the generating function.
Examples from Banderier and Flajolet deal with constrained walks ("meanders" and "excursions") while this sequence counts unrestricted paths.
Logarithmic derivative of A187430 (when offset 1). - Paul D. Hanna, May 31 2015
LINKS
C. Banderier and P. Flajolet, Basic analytic combinatorics of directed lattice paths, Theoretical Computer Science, Vol. 281:1-2, pp. 37-80, 2002.
K. Lakatos-Lindenberg and K. E. Shuler, Random walks with nonnearest neighbor transitions. I. Analytic 1-D theory for next-nearest neighbor and exponentially distributed steps, Journal of Mathematical Physics, Vol. 12 Num.4, pp. 633-652, 1971.
Eric Weisstein's World of Mathematics, Eight Curve
FORMULA
G.f. in Maple notation: {x*(1+6*x)*(1-4*x)*(4+9*x)*diff(G(x), x, x)=2*(270*x^3+84*x^2+13*x-1)*diff(G(x), x)+4*x*(12+27*x)*G(x), G(0)=1, D(G)(0)=0} rec; 2*(n+1)*(2*n+1)*a(n+1)+n*(17*n-43)*a(n)=(78*n^2-66*n+36)*a(n-1)+(216*n^2-540*n+324)*a(n-2).
GFun gives the following algebraic equation for generating function: x+2*(1-4*x)*(3*x-2)*g(x)^2+(1-4*x)^2*(9*x+4)*g(x)^4=0. - Sergey Perepechko, Sep 06 2004
a(n) = (2^(2n+1) / Pi) * Integral(cos(t)^n*cos(3*t)^n, t=0..Pi/2); a(n) = Sum_{k=0..n} binomial(n,k)*binomial(4*n-2*k,2*n-k)*(-3)^k. G.f.: (1 + sqrt(1-4*x)) / ( sqrt(1-4*x) * ( sqrt(1+6*x+2*sqrt(9*x^2+4*x)) + sqrt(1+6*x-2*sqrt(9*x^2+4*x)) ) ). - Max Alekseyev, Apr 19 2006
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n,2*n-3*k). - Max Alekseyev, Feb 08 2008
a(n) = Sum_{k=0..2n} (-1)^k*binomial(2n,k)*A027907(n,k) where A027907 is the triangle of trinomial coefficients. - Paul D. Hanna, Nov 30 2009
a(n) = ((n-1)*(35*n^2-49*n+12) *a(n-1) +18*(n-1)*(2*n-3)*(5*n-2) *a(n-2)) / (2*n*(2*n-1)*(5*n-7)) for n>=2, a(n) = 1-n for n<2. - Alois P. Heinz, May 20 2013
a(n) ~ 4^n / sqrt(5*Pi*n). - Vaclav Kotesovec, Sep 12 2014
a(n) is the coefficient of x^(2*n) in ((1-x)*(1-x^3))^n. - Max Alekseyev, Jun 01 2015
a(n) = (-1)^n*binomial(2*n,n)*hypergeom([-n,n/2,(n+1)/2],[n,n+1],4). - Peter Luschny, Nov 02 2016
From Peter Bala, Feb 08 2022: (Start)
a(n) = Sum_{k = 0..n} (-1)^k*binomial(2*n,k)*binomial(3*n-2*k-1,n-k).
a(n) = Sum_{k = 0..floor(n/2)} binomial(2*n,k)*binomial(n-k-1,n-2*k).
a(n) = [x^n] ((1 - x + x^2)/(1 - x))^(2*n).
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for any prime p and positive integers n and k.
Conjecture: the stronger congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(2*k) ) hold for any prime p, except p = 3, and positive integers n and k.(End)
EXAMPLE
a(3)=6 because 0=+2-1-1, 0=-2+1+1, 0=-1-1+2, 0=+1+1-2, 0=+1-2+1, 0=-1+2-1.
MAPLE
a:=array(0..20):a[0]:=1:a[1]:=0:a[2]:=4:for n from 2 to 19 do a[n+1]:=(-n*(17*n-43)*a[n]+(78*n^2-66*n+36)*a[n-1]+(216*n^2-540*n+324)*a[n-2])/(2*(n+1)*(2*n+1)):print(n+1, a[n+1]) od:
seq(coeff( (t^2+t+1/t+1/t^2)^n, t, 0), n=0..24); # Mark van Hoeij, May 20 2013
MATHEMATICA
a[n_] := Binomial[4n, 2n]*Hypergeometric2F1[-2n, -n, 1/2 - 2n, 3/4]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Nov 22 2012 *)
PROG
(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n-2*k, 2*n-k)*(-3)^k) /* Max Alekseyev, Apr 19 2006 */
(PARI) a(n)=sum(k=0, n, binomial(n, k)*binomial(n, 2*n-3*k)) /* Max Alekseyev, Feb 08 2008 */
(PARI) a(n)=sum(k=0, 2*n, (-1)^k*binomial(2*n, k)*polcoeff((1+x+x^2)^n, k)) /* Paul D. Hanna, Nov 30 2009 */
(PARI) a(n) = polcoeff(( (1-x)*(1-x^3) + O(x^(2*n+1)) )^n, 2*n); /* Max Alekseyev, Jun 01 2015 */
CROSSREFS
Sequence in context: A175061 A222502 A092187 * A056315 A103234 A074061
KEYWORD
nonn,easy
AUTHOR
Sergey Perepechko, Apr 19 2004
EXTENSIONS
More terms from Max Alekseyev, Apr 19 2006
STATUS
approved

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Last modified April 20 02:10 EDT 2024. Contains 371798 sequences. (Running on oeis4.)