

A286928


Number of solutions to the equation x_1 + ... + x_n =0 satisfying n<=x_i<=n (1<=i<=n).


3



1, 5, 37, 489, 8801, 204763, 5832765, 197018321, 7702189345, 342237634221, 17039997700639, 939906923598525, 56899727331724863, 3751071253402671045, 267515957818316650221, 20522595752454270972321, 1685273102403664075044305, 147501996974331775160471677
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OFFSET

1,2


COMMENTS

The number of variables in the equation is exactly n and each variable can have a value of n to n.
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.


LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..100


FORMULA

a(n) = Sum_{i=0..floor(n/2)} (1)^i*binomial(n*(n+1)i*(2*n+1)1, n1)*binomial(n, i).


EXAMPLE

Case n=3:
Solutions are: {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 6+6+6+6+3+3+6+1 = 37 solutions so a(3)=37.


MATHEMATICA

a[n_] := Sum[(1)^i*Binomial[n, i]*Binomial[n*(n+1)  i*(2n+1)  1, n1], {i, 0, n/2}]; Array[a, 18] (* JeanFrançois Alcover, Oct 01 2017 *)


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)=compositions(n*(n+1), 2*n+1, n);


CROSSREFS

Cf. A160492, A208597.
Sequence in context: A318002 A304865 A003709 * A321042 A244820 A246534
Adjacent sequences: A286925 A286926 A286927 * A286929 A286930 A286931


KEYWORD

nonn


AUTHOR

Andrew Howroyd, May 16 2017


STATUS

approved



