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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.

%I A201805 #100 Apr 20 2025 03:48:42
%S A201805 1,1,5,13,61,221,1001,4145,18733,82381,375745,1703945,7858225,
%T A201805 36279985,168992045,789433013,3707816333,17467638925,82599195809,
%U A201805 391645961993,1862242702201,8875355178521,42394598106965,202903189757053
%N 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.
%C A201805 Column 2 of A201811.
%C A201805 Also the number of walks of length n from a vertex to itself on the infinite square lattice with a self loop on each vertex. - _Pierre-Louis Giscard_, Jun 25 2014
%C A201805 Also the number of 3D walks of length n in a half-space returning to axis of origin. - _Nachum Dershowitz_, Aug 04 2020
%C A201805 The central column of a number pyramid P(j,k,m), where P(j,k,m) = P(j,k,m-1) + P(j-1,k,m-1) + P(j+1,k,m-1) + P(j,k-1,m-1) + P(j,k+1,m-1). P(1,1,1) = 1. j, k = 1..2*m+1. m >=1. - _Yuriy Sibirmovsky_, Sep 17 2016
%C A201805 Row sums of A282252. - _Peter Bala_, Feb 12 2017
%H A201805 R. H. Hardin and Seiichi Manyama, <a href="/A201805/b201805.txt">Table of n, a(n) for n = 0..1000</a> (a(1)-a(210) from R. H. Hardin)
%H A201805 Nachum Dershowitz, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Dershowitz/dersh3.html">Touchard's Drunkard</a>, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5. The sequence is type bbd in Table 3.
%H A201805 Yuriy Sibirmovsky, <a href="/A201805/a201805.png">a(n) as the central column of a number pyramid (zeros are left blank)</a>.
%F A201805 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
%F A201805 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
%F A201805 From _Pierre-Louis Giscard_, Jun 25 2014 : (Start)
%F A201805 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.
%F A201805 a(n) = 3F2([1/2, 1/2 - n/2, -n/2], [1, 1], 16).
%F A201805 a(n) = Sum_{k=0..n} C(n,2k)*C(2k,k)^2.
%F A201805 O.g.f.: 2F1([1/2, 1/2], [1], 16*x^2/(1-x)^2)/(1-x) with 2F1 the Hypergeometric function.
%F A201805 E.g.f.: e^x I_{0}(2x)^2 with I_a(x) the modified Bessel function I of the first kind. (End)
%F A201805 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
%F A201805 a(n) ~ 5^(n+1)/(4*Pi*n). - _Vaclav Kotesovec_, Oct 03 2016
%e A201805 Some solutions for n=9
%e A201805 .-1...-1....1....1....0...-2....2...-1...-2...-2....1....1....1....2....0....1
%e A201805 ..1...-2...-2...-2...-1...-2....1....0....2....1....0...-2...-1...-2....0...-1
%e A201805 ..0....0....2....1...-1....2...-1....1....0...-2...-1....1...-2....1...-1....1
%e A201805 .-1...-2....2....0...-2....1....0....2....0....0...-1...-1....2...-1....0....1
%e A201805 ..2....1....0....2...-1....0....1...-2...-1...-1....1....0...-2....1....0...-1
%e A201805 ..0....2...-2...-1....2....0...-2...-2....0....2....1...-1...-2....2....2....1
%e A201805 ..1....1...-2....1....1...-1....0....2....1...-2....0....2....2...-2...-2...-1
%e A201805 ..0...-1....2...-1....1....2...-1...-2....1....2...-1...-2....0....0....0....0
%e A201805 .-2....2...-1...-1....1....0....0....2...-1....2....0....2....2...-1....1...-1
%t A201805 a[n_]=HypergeometricPFQ[{1/2, 1/2 - n/2, -(n/2)}, {1, 1}, 16]; (* or *)
%t A201805 a[n_]=Sum[Binomial[n, 2 k] Binomial[2 k, k]^2, {k, 0, n}]; (* or *)
%t A201805 Hypergeometric2F1[1/2, 1/2, 1, 16*x^2/(1 - x)^2]/(1 - x); (* O.g.f. *)
%t A201805 Exp[x] BesselI[0, 2 x] BesselI[0, 2 x]; (* E.g.f. *)(* _Pierre-Louis Giscard_, Jun 25 2014 *)
%t A201805 Nm=100;
%t A201805 C1=Table[0,{j,1,Nm},{k,1,Nm}];
%t A201805 C1[[Nm/2,Nm/2]]=1;
%t A201805 C2=C1;
%t A201805 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]]];
%t A201805 C1=C2,{n,1,20}] (* _Yuriy Sibirmovsky_, Sep 17 2016 *)
%o A201805 (PARI) a(n) = sum(k=0, n, binomial(n, 2*k)*binomial(2*k,k)^2); \\ _Michel Marcus_, Jun 25 2014
%o A201805 (PARI) {a(n)=polcoeff(1/agm(1+3*x, 1-5*x +x*O(x^n)), n)}
%o A201805 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Aug 31 2014
%o A201805 (PARI) {a(n) = polcoef(polcoef((1+x+y+1/x+1/y)^n, 0), 0)} \\ _Seiichi Manyama_, Oct 26 2019
%Y A201805 Sum_{k=0..n} C(n,2k)*C(2k,k)^m: A002426 (m=1), this sequence (m=2).
%Y A201805 Cf. A202814, A282252.
%K A201805 nonn,easy
%O A201805 0,3
%A A201805 _R. H. Hardin_, Dec 05 2011
%E A201805 a(0)=1 prepended by _Seiichi Manyama_, Dec 02 2016