A201554 Number of arrays of 7 integers in -n..n with sum zero.

1, 393, 8135, 60691, 273127, 908755, 2473325, 5832765, 12354469, 24072133, 43874139, 75715487, 124853275, 198105727, 304134769, 453752153, 660249129, 939749665, 1311587215, 1798705035, 2428080047, 3231170251, 4244385685, 5509582933

Offset

0,2

Comments

Row 7 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]

Formula

Empirical: a(n) = 1+ 7*n*(n+1)*(841*n^4+1682*n^3+1568*n^2+727*n+222)/180.

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

G.f.: (393 + 5384*x + 11999*x^2 + 5370*x^3 + 407*x^4 - 6*x^5 + x^6) / (1 - x)^7. -

a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.

(End)

a(n) = [x^(7*n)] (Sum_{k=0..2*n} x^k)^7. - Seiichi Manyama, Dec 14 2018

Example

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

Mathematica

a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^7, x], x, 7n]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *)

Prog

(PARI) {a(n) = polcoeff((sum(k=0, 2*n, x^k))^7, 7*n, x)} \\ Seiichi Manyama, Dec 14 2018

Crossrefs

Cf. A201552.

Sequence in context: A047825 A227942 A201815 * A172636 A202046 A172737

Adjacent sequences: A201551 A201552 A201553 * A201555 A201556 A201557

Keyword

nonn

Author

R. H. Hardin, Dec 02 2011

Extensions

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

Status

approved