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
Keywords
Examples
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
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]
- Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).
Crossrefs
Cf. A201552.
Programs
-
Mathematica
a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^7, x], x, 7n]; Array[a, 25, 0] (* Amiram Eldar, Dec 14 2018 *) -
PARI
{a(n) = polcoeff((sum(k=0, 2*n, x^k))^7, 7*n, x)} \\ Seiichi Manyama, Dec 14 2018
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
Barker conjectures confirmed using technique similar to A201553.
Extensions
a(0)=1 prepended by Seiichi Manyama, Dec 14 2018
Comments