A201805 Number of arrays of n integers in -2..2 with sum zero and equal numbers of elements greater than zero and less than zero.
1, 1, 5, 13, 61, 221, 1001, 4145, 18733, 82381, 375745, 1703945, 7858225, 36279985, 168992045, 789433013, 3707816333, 17467638925, 82599195809, 391645961993, 1862242702201, 8875355178521, 42394598106965, 202903189757053
Offset: 0
Examples
Some solutions for n=9 .-1...-1....1....1....0...-2....2...-1...-2...-2....1....1....1....2....0....1 ..1...-2...-2...-2...-1...-2....1....0....2....1....0...-2...-1...-2....0...-1 ..0....0....2....1...-1....2...-1....1....0...-2...-1....1...-2....1...-1....1 .-1...-2....2....0...-2....1....0....2....0....0...-1...-1....2...-1....0....1 ..2....1....0....2...-1....0....1...-2...-1...-1....1....0...-2....1....0...-1 ..0....2...-2...-1....2....0...-2...-2....0....2....1...-1...-2....2....2....1 ..1....1...-2....1....1...-1....0....2....1...-2....0....2....2...-2...-2...-1 ..0...-1....2...-1....1....2...-1...-2....1....2...-1...-2....0....0....0....0 .-2....2...-1...-1....1....0....0....2...-1....2....0....2....2...-1....1...-1
Links
- R. H. Hardin and Seiichi Manyama, Table of n, a(n) for n = 0..1000 (a(1)-a(210) from R. H. Hardin)
- Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5. The sequence is type bbd in Table 3.
- Yuriy Sibirmovsky, a(n) as the central column of a number pyramid (zeros are left blank).
Programs
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Mathematica
a[n_]=HypergeometricPFQ[{1/2, 1/2 - n/2, -(n/2)}, {1, 1}, 16]; (* or *) a[n_]=Sum[Binomial[n, 2 k] Binomial[2 k, k]^2, {k, 0, n}]; (* or *) Hypergeometric2F1[1/2, 1/2, 1, 16*x^2/(1 - x)^2]/(1 - x); (* O.g.f. *) Exp[x] BesselI[0, 2 x] BesselI[0, 2 x]; (* E.g.f. *)(* Pierre-Louis Giscard, Jun 25 2014 *) Nm=100; C1=Table[0,{j,1,Nm},{k,1,Nm}]; C1[[Nm/2,Nm/2]]=1; C2=C1; Do[Do[C2[[j,k]]=C1[[j-1,k]]+C1[[j+1,k]]+C1[[j,k-1]]+C1[[j,k+1]]+C1[[j,k]],{j,2,Nm-1},{k,2,Nm-1}];Print[n," ",C2[[Nm/2,Nm/2]]]; C1=C2,{n,1,20}] (* Yuriy Sibirmovsky, Sep 17 2016 *) -
PARI
a(n) = sum(k=0, n, binomial(n, 2*k)*binomial(2*k,k)^2); \\ Michel Marcus, Jun 25 2014
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PARI
{a(n)=polcoeff(1/agm(1+3*x, 1-5*x +x*O(x^n)), n)} for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 31 2014 -
PARI
{a(n) = polcoef(polcoef((1+x+y+1/x+1/y)^n, 0), 0)} \\ Seiichi Manyama, Oct 26 2019
Formula
Empirical: n^2*a(n) = (3*n^2-3*n+1)*a(n-1) + 13*(n-1)^2*a(n-2) - 15*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Oct 19 2012
a(n) appears to be the constant term of (1 + X + 1/X + Y + 1/Y)^n, which has o.g.f. hypergeom([1/2, 1/2],[1],16*x^2/(1-x)^2)/(1-x). - Mark van Hoeij, May 07 2013
From Pierre-Louis Giscard, Jun 25 2014 : (Start)
a(n) is exactly the constant term of (1 + X + 1/X + Y + 1/Y)^n since this generates closed walks on the square lattice with self-loops. Non-constant terms generate walks to the neighbors of a vertex. Removing the 1 is equivalent to removing the self-loops.
a(n) = 3F2([1/2, 1/2 - n/2, -n/2], [1, 1], 16).
a(n) = Sum_{k=0..n} C(n,2k)*C(2k,k)^2.
O.g.f.: 2F1([1/2, 1/2], [1], 16*x^2/(1-x)^2)/(1-x) with 2F1 the Hypergeometric function.
E.g.f.: e^x I_{0}(2x)^2 with I_a(x) the modified Bessel function I of the first kind. (End)
O.g.f.: 1 / AGM(1+3*x, 1-5*x), given a(0)=1, where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean. - Paul D. Hanna, Aug 31 2014
a(n) ~ 5^(n+1)/(4*Pi*n). - Vaclav Kotesovec, Oct 03 2016
Extensions
a(0)=1 prepended by Seiichi Manyama, Dec 02 2016
Comments