

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, k1)*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((rk)/m), (1)^i*binomial(r1i*m, k1)*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 * A318017 A273091 A086721
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



