%I #12 Dec 07 2019 12:18:26

%S 1,6,45,560,9795,223524,6284089,210208560,8156750283,360297117070,

%T 17853149451841,980844453593160,59179098916735213,3890176308574524934,

%U 276750779199166606705,21185250061147839785120,1736385140876356212244563,151719500906542020597450498

%N a(n) = number of solutions to an equation x_1 + ... + x_j =0 with 1<=j<=n satisfying -n<=x_i<=n (1<=i<=j).

%C The number of variables in the equation can be from 1 to n and each variable can have a value of -n to n. See A286928 for the case of exactly n variables. - _Andrew Howroyd_, May 16 2017

%F a(n) = Sum_{k=1..n} Sum_{i=0..floor(k/2)} (-1)^i*binomial(k*(n+1)-i*(2*n+1)-1, k-1)*binomial(k, i). - _Andrew Howroyd_, May 16 2017

%e From _Andrew Howroyd_, May 16 2017 (Start)

%e Case n=3:

%e 1 variable: {0} is only solution.

%e 2 variables: {-3,3}, {-2,2}, {-1,1}, {0,0}, {1,-1}, {2,-2}, {3,-3}.

%e 3 variables: {-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 1 + 7 + 37 = 45 solutions so a(3)=45.

%e (End)

%t zerocompositionswithzero[p_] := Module[{united = {}, i, zerosums = {}, count = 0}, For[i = 1, i <= p, i = i + 1, united = Union[united, Tuples[Table[x, {x, -p, p}], i]] ]; For[i = 1, i <= Length[united], i = i + 1, If[Sum[united[[i, j]], {j, 1, Length[united[[i]]]}] == 0, zerosums = Append[zerosums, united[[i]]]; count = count + 1;]; ]; Return[{count, zerosums}]; ];

%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)=sum(v=1,n,compositions(v*(n+1),2*n+1,v)); \\ _Andrew Howroyd_, May 16 2017

%o (Python)

%o from sympy import binomial

%o def C(r, m, k): return sum([(-1)**i*binomial(r - 1 - i*m, k - 1)*binomial(k, i) for i in range(int((r - k)/m) + 1)])

%o def a(n): return sum([C(v*(n + 1), 2*n + 1, v) for v in range(1, n + 1)]) # _Indranil Ghosh_, May 16 2017, after the PARI program by _Andrew Howroyd_

%Y Cf. A286928.

%K nonn

%O 1,2

%A _Srikanth K S_, May 15 2009

%E Name clarified and a(6)-a(18) from _Andrew Howroyd_, May 16 2017