A201550 - OEIS

%I #36 Oct 16 2024 09:23:38

%S 1,1,13,127,1469,17151,204763,2473325,30162301,370487485,4577127763,

%T 56813989827,707972099627,8851373201919,110976634957761,

%U 1394804756117877,17567994350713469,221690794842728445,2802194053806820153

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

%C Also largest coefficient of (1+x+...+x^12)^n. - _Vaclav Kotesovec_, Aug 09 2013

%H Seiichi Manyama, <a href="/A201550/b201550.txt">Table of n, a(n) for n = 0..450</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 Vaclav Kotesovec, <a href="/A201550/a201550.txt">Recurrence</a>

%F a(n) ~ 13^n / sqrt(28*Pi*n). - _Vaclav Kotesovec_, Aug 09 2013

%F a(n) = Sum_{k = 0..floor(n/2)} (-1)^k * binomial(n, k)*binomial(7*n-13*k-1, n-1). - _Peter Bala_, Oct 16 2024

%e Some solutions for n=5

%e .-2...-5...-2...-1....3...-6....0...-3....1....6...-6...-2....5....0...-4...-3

%e ..2...-3...-2....3....0...-1....6...-4....6....1....5....2...-1....3....2....3

%e ..0...-4....4...-6...-4....1...-3....0...-4...-5....0...-6...-3....0....4...-4

%e ..0....6....3....5...-5....6....0....4....3...-4....4....0...-5...-3....3...-1

%e ..0....6...-3...-1....6....0...-3....3...-6....2...-3....6....4....0...-5....5

%p seq(add((-1)^k * binomial(n, k)*binomial(7*n-13*k-1, n-1), k = 0..floor(n/2)), n = 0..20); # _Peter Bala_, Oct 16 2024

%t Table[Coefficient[Expand[Sum[x^j,{j,0,12}]^n],x^(6*n)],{n,1,20}] (* _Vaclav Kotesovec_, Aug 09 2013 *)

%o (PARI) {a(n) = polcoeff((sum(k=0, 12, x^k))^n, 6*n, x)} \\ _Seiichi Manyama_, Dec 14 2018

%Y Column 6 of A201552.

%Y Cf. A001405, A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015, A201549, A225779.

%K nonn,easy

%O 0,3

%A _R. H. Hardin_, Dec 02 2011

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