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%I #32 Jan 19 2019 16:32:15
%S 1,1,5,37,489,8801,204763,5832765,197018321,7702189345,342237634221,
%T 17039997700639,939906923598525,56899727331724863,3751071253402671045,
%U 267515957818316650221,20522595752454270972321,1685273102403664075044305,147501996974331775160471677
%N Number of solutions to the equation x_1 + ... + x_n =0 satisfying -n<=x_i<=n (1<=i<=n).
%C The number of variables in the equation is exactly n and each variable can have a value of -n to n.
%C Also the number of compositions of n^2 into a maximum of n parts and each part having a maximum value of 2n. Equivalently, the number of compositions of n(n+1) into exactly n parts and each part having a maximum value of 2n+1.
%H Seiichi Manyama, <a href="/A286928/b286928.txt">Table of n, a(n) for n = 0..352</a> (terms 1..100 from Andrew Howroyd) (It was suggested that the initial terms of this b-file were wrong, but in fact they are correct. - _Vaclav Kotesovec_, Jan 19 2019)
%F a(n) = Sum_{i=0..floor(n/2)} (-1)^i*binomial(n*(n+1)-i*(2*n+1)-1, n-1)*binomial(n, i).
%F a(n) = [x^(n^2)] (Sum_{k=0..2*n} x^k)^n. - _Seiichi Manyama_, Dec 13 2018
%F a(n) ~ sqrt(3) * exp(1/2) * 2^(n - 1/2) * n^(n - 3/2) / sqrt(Pi). - _Vaclav Kotesovec_, Dec 15 2018
%e Case n=3:
%e Solutions are: {-3 0 3}x6, {-3 1 2}x6, {-2 -1 3}x6, {-2 0 2}x6,
%e {-2 1 1}x3, {-1 -1 2}x3, {-1 0 1}x6, {0 0 0}x1
%e In the above, {-3 0 3}x6 means that the values can be expanded to 6 solutions by considering different orderings.
%e In total there are 6+6+6+6+3+3+6+1 = 37 solutions so a(3)=37.
%t a[n_] := Sum[(-1)^i*Binomial[n, i]*Binomial[n*(n+1) - i*(2n+1) - 1, n-1], {i, 0, n/2}]; Array[a, 18] (* _Jean-François Alcover_, Oct 01 2017 *)
%o (PARI)
%o \\ nr compositions of r with max value m into exactly k parts
%o compositions(r,m,k)=sum(i=0,floor((r-k)/m),(-1)^i*binomial(r-1-i*m, k-1)*binomial(k, i));
%o a(n)=compositions(n*(n+1),2*n+1,n);
%o (PARI) {a(n) = polcoeff((sum(k=0, 2*n, x^k))^n, n^2, x)} \\ _Seiichi Manyama_, Dec 13 2018
%Y Cf. A160492, A208597.
%K nonn
%O 0,3
%A _Andrew Howroyd_, May 16 2017
%E a(0)=1 prepended by _Seiichi Manyama_, Dec 13 2018
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