A286928 Number of solutions to the equation x_1 + ... + x_n =0 satisfying -n<=x_i<=n (1<=i<=n).

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

Offset

0,3

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

Seiichi Manyama, Table of n, a(n) for n = 0..352 (terms 1..100 from Andrew Howroyd) (It was suggested that the initial terms of this b-file were wrong, but in fact they are correct. - Vaclav Kotesovec, Jan 19 2019)

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).

a(n) = [x^(n^2)] (Sum_{k=0..2*n} x^k)^n. - Seiichi Manyama, Dec 13 2018

a(n) ~ sqrt(3) * exp(1/2) * 2^(n - 1/2) * n^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Dec 15 2018

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-François 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);
(PARI) {a(n) = polcoeff((sum(k=0, 2*n, x^k))^n, n^2, x)} \\ Seiichi Manyama, Dec 13 2018

Crossrefs

Cf. A160492, A208597.

Sequence in context: A304865 A003709 A361281 * A321042 A244820 A359796

Adjacent sequences: A286925 A286926 A286927 * A286929 A286930 A286931

Keyword

nonn

Author

Andrew Howroyd, May 16 2017

Extensions

a(0)=1 prepended by Seiichi Manyama, Dec 13 2018

Status

approved