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%I #22 Jan 19 2019 12:38:17
%S 1,393,8135,60691,273127,908755,2473325,5832765,12354469,24072133,
%T 43874139,75715487,124853275,198105727,304134769,453752153,660249129,
%U 939749665,1311587215,1798705035,2428080047,3231170251,4244385685,5509582933
%N Number of arrays of 7 integers in -n..n with sum zero.
%C Row 7 of A201552.
%H Seiichi Manyama, <a href="/A201554/b201554.txt">Table of n, a(n) for n = 0..10000</a> (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]
%F Empirical: a(n) = 1+ 7*n*(n+1)*(841*n^4+1682*n^3+1568*n^2+727*n+222)/180.
%F Conjectures from _Colin Barker_, May 23 2018: (Start)
%F G.f.: (393 + 5384*x + 11999*x^2 + 5370*x^3 + 407*x^4 - 6*x^5 + x^6) / (1 - x)^7. -
%F 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.
%F (End)
%F a(n) = [x^(7*n)] (Sum_{k=0..2*n} x^k)^7. - _Seiichi Manyama_, Dec 14 2018
%e Some solutions for n=3:
%e ..1....3....2....2....3...-2....0...-1...-2....1....0....1...-2...-3....1...-1
%e ..3....2...-3....0...-2...-2....1...-3....1...-2....2....2....3....1....2...-1
%e .-3...-3....3....2...-2....1...-1....3...-3....3....1....1....0....0...-1....3
%e .-3...-2....2...-3....0....1....2....2...-1....1...-2...-3...-1....3....0....3
%e ..0....0...-1....3...-1....1....2....1....1....1....1....2...-2...-1....0....2
%e .-1....0...-1...-1....2....3...-1...-1....1...-1...-2...-1....2....2...-2...-3
%e ..3....0...-2...-3....0...-2...-3...-1....3...-3....0...-2....0...-2....0...-3
%t a[n_] := Coefficient[Expand[Sum[x^k, {k, 0, 2n}]^7, x], x, 7n]; Array[a, 25, 0] (* _Amiram Eldar_, Dec 14 2018 *)
%o (PARI) {a(n) = polcoeff((sum(k=0, 2*n, x^k))^7, 7*n, x)} \\ _Seiichi Manyama_, Dec 14 2018
%Y Cf. A201552.
%K nonn
%O 0,2
%A _R. H. Hardin_, Dec 02 2011
%E a(0)=1 prepended by _Seiichi Manyama_, Dec 14 2018
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