A201553 Number of arrays of 6 integers in -n..n with sum zero.

1, 141, 1751, 9331, 32661, 88913, 204763, 418503, 782153, 1363573, 2248575, 3543035, 5375005, 7896825, 11287235, 15753487, 21533457, 28897757, 38151847, 49638147, 63738149, 80874529, 101513259, 126165719, 155390809, 189797061, 230044751

Offset

0,2

Comments

Row 6 of A201552.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 1..210 from R. H. Hardin) [It was suggested that the initial terms of this b-file were wrong, but in fact they are correct. - N. J. A. Sloane, Jan 19 2019]

Robert Israel, Proof of "empirical" formula

Formula

Empirical: a(n) = (2*n+1)*(44*n^4+88*n^3+71*n^2+27*n+5)/5.

Empirical formula verified (see link) by Robert Israel, Dec 14 2018.

Empirical: a(n)= integral( (sin((n+1/2)x)/sin(x/2))^6, x=0..Pi)/Pi. - Yalcin Aktar, Dec 03 2011

Conjectures from Colin Barker, May 23 2018: (Start)

G.f.: x*(141 + 905*x + 940*x^2 + 120*x^3 + 7*x^4 - x^5) / (1 - x)^6.

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

(End)

a(n) = [x^(6*n)] (Sum_{k=0..2*n} x^k)^6. - Seiichi Manyama, Dec 14 2018

Example

Some solutions for n=5:
..4....5....4...-2...-4....5...-1...-2...-1...-3...-3....0....2...-4....2...-5
..1...-4....5....3....4...-4....1....1....1....0....2...-2....1...-2...-1....1
.-2....3...-5....3....1....0...-4....2...-2....3....3....0....4....3....4....3
.-3...-3...-4....2....2...-3....5....4....4....0...-2....2....0....4...-1...-2
..5....4...-4...-2...-3...-1...-4...-1....1....0...-2....3...-4...-5...-2....4
.-5...-5....4...-4....0....3....3...-4...-3....0....2...-3...-3....4...-2...-1

Mathematica

a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^6, x], x, 6n]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

Prog

(PARI) {a(n) = polcoeff((sum(k=0, 2*n, x^k))^6, 6*n, x)} \\ Seiichi Manyama, Dec 14 2018

Crossrefs

Cf. A201552.

Sequence in context: A201814 A235543 A204469 * A281560 A186955 A005849

Adjacent sequences: A201550 A201551 A201552 * A201554 A201555 A201556

Keyword

nonn

Author

R. H. Hardin, Dec 02 2011

Extensions

a(0)=1 prepended by Seiichi Manyama, Dec 14 2018

Status

approved