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A160492 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). 1
1, 6, 45, 560, 9795, 223524, 6284089, 210208560, 8156750283, 360297117070, 17853149451841, 980844453593160, 59179098916735213, 3890176308574524934, 276750779199166606705, 21185250061147839785120, 1736385140876356212244563, 151719500906542020597450498 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..18.

FORMULA

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

EXAMPLE

From Andrew Howroyd, May 16 2017 (Start)

Case n=3:

1 variable: {0} is only solution.

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

3 variables: {-3 0 3}x6, {-3 1 2}x6, {-2 -1 3}x6, {-2 0 2}x6,

             {-2 1 1}x3, {-1 -1 2}x3, {-1 0 1}x6, {0 0 0}x1

In the above, {-3 0 3}x6 means that the values can be expanded to 6 solutions by considering different orderings.

In total there are 1 + 7 + 37 = 45 solutions so a(3)=45.

(End)

MATHEMATICA

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}]; ];

PROG

(PARI)

\\ nr compositions of r with max value m into exactly k parts.

compositions(r, m, k)=sum(i=0, floor((r-k)/m), (-1)^i*binomial(r-1-i*m, k-1)*binomial(k, i));

a(n)=sum(v=1, n, compositions(v*(n+1), 2*n+1, v));  \\ Andrew Howroyd, May 16 2017

(Python)

from sympy import binomial

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

def a(n): return sum([C(v*(n + 1), 2*n + 1, v) for v in xrange(1, n + 1)]) # Indranil Ghosh, May 16 2017, after the PARI program by Andrew Howroyd

CROSSREFS

Cf. A286928.

Sequence in context: A109516 A245493 A078865 * A273091 A086721 A145002

Adjacent sequences:  A160489 A160490 A160491 * A160493 A160494 A160495

KEYWORD

nonn

AUTHOR

Srikanth K S, May 15 2009

EXTENSIONS

Name clarified and a(6)-a(18) from Andrew Howroyd, May 16 2017

STATUS

approved

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Last modified December 17 04:09 EST 2017. Contains 296096 sequences.