

A286928


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


7



1, 1, 5, 37, 489, 8801, 204763, 5832765, 197018321, 7702189345, 342237634221, 17039997700639, 939906923598525, 56899727331724863, 3751071253402671045, 267515957818316650221, 20522595752454270972321, 1685273102403664075044305, 147501996974331775160471677
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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 bfile 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, n1)*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, 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);
(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: A323567 A304865 A003709 * A321042 A244820 A246534
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



