login
Number of arrays of 6 integers in -n..n with sum zero.
3

%I #39 Sep 29 2024 03:10:24

%S 1,141,1751,9331,32661,88913,204763,418503,782153,1363573,2248575,

%T 3543035,5375005,7896825,11287235,15753487,21533457,28897757,38151847,

%U 49638147,63738149,80874529,101513259,126165719,155390809,189797061,230044751

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

%C Row 6 of A201552.

%H Seiichi Manyama, <a href="/A201553/b201553.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]

%H Robert Israel, <a href="/A201553/a201553.pdf">Proof of "empirical" formula</a>.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

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

%F Empirical formula verified (see link) by _Robert Israel_, Dec 14 2018.

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

%F Conjectures from _Colin Barker_, May 23 2018: (Start)

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

%F 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).

%F (End)

%F a(n) = [x^(6*n)] (Sum_{k=0..2*n} x^k)^6. - _Seiichi Manyama_, Dec 14 2018

%F E.g.f.: exp(x)*(5 + 700*x + 3675*x^2 + 3750*x^3 + 1100*x^4 + 88*x^5)/5. - _Stefano Spezia_, Sep 28 2024

%e Some solutions for n=5:

%e ..4....5....4...-2...-4....5...-1...-2...-1...-3...-3....0....2...-4....2...-5

%e ..1...-4....5....3....4...-4....1....1....1....0....2...-2....1...-2...-1....1

%e .-2....3...-5....3....1....0...-4....2...-2....3....3....0....4....3....4....3

%e .-3...-3...-4....2....2...-3....5....4....4....0...-2....2....0....4...-1...-2

%e ..5....4...-4...-2...-3...-1...-4...-1....1....0...-2....3...-4...-5...-2....4

%e .-5...-5....4...-4....0....3....3...-4...-3....0....2...-3...-3....4...-2...-1

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

%o (PARI) {a(n) = polcoeff((sum(k=0, 2*n, x^k))^6, 6*n, x)} \\ _Seiichi Manyama_, Dec 14 2018

%Y Cf. A201552.

%K nonn,easy

%O 0,2

%A _R. H. Hardin_, Dec 02 2011

%E a(0)=1 prepended by _Seiichi Manyama_, Dec 14 2018