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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 (list; graph; refs; listen; history; text; internal format)
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, n-1)*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, n-1], {i, 0, n/2}]; Array[a, 18] (* Jean-Fran├žois 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((r-k)/m), (-1)^i*binomial(r-1-i*m, k-1)*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

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Last modified December 12 11:34 EST 2018. Contains 318060 sequences. (Running on oeis4.)