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A201553 Number of arrays of 6 integers in -n..n with sum zero. 2
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 (list; graph; refs; listen; history; text; internal format)
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)

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

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)