A201549 - OEIS

%I #29 Oct 16 2024 09:23:08

%S 1,1,11,91,891,8801,88913,908755,9377467,97464799,1018872811,

%T 10701243741,112835748609,1193692544825,12663809507129,

%U 134678108144591,1435345208419771,15326122342137035,163920458145421109

%N Number of arrays of n integers in -5..5 with sum zero.

%C Also largest coefficient of (1+x+...+x^10)^n. - _Vaclav Kotesovec_, Aug 09 2013

%H Seiichi Manyama, <a href="/A201549/b201549.txt">Table of n, a(n) for n = 0..500</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 Recurrence: 726*(n-2) * (n-1) * (2*n-1) * (3*n-7) * (3*n-1) * (5*n-19) * (5*n-9) * (5*n-8) * (6*n-25) * (6*n-13) * (6*n-1) * (2280*n^4  -  18164*n^3  +  49523*n^2  -  53013*n  +  18900)*a(n-3)  -  3*(2*n-1) * (3*n-7) * (3*n-1) * (5*n-19) * (5*n-14) * (5*n-9) * (5*n-8) * (6*n-25) * (6*n-19) * (6*n-13) * (6*n-1) * (4651*n^4  -  18604*n^3  +  27451*n^2  -  17694*n  +  4200)*a(n-1)  +  3993*(n-3) * (n-2) * (n-1) * (3*n-4) * (5*n-14) * (5*n-4) * (5*n-3) * (6*n-19) * (6*n-7) * (6*n-1) * (5310*n^5  -  65313*n^4  +  295326*n^3  -  594091*n^2  +  499480*n  -  112320)*a(n-4)  -  33*(n-1) * (3*n-4) * (5*n-19) * (5*n-14) * (5*n-4) * (5*n-3) * (6*n-25) * (6*n-19) * (6*n-7) * (45306*n^6  -  385101*n^5  +  1267841*n^4  -  2002349*n^3  +  1504595*n^2  -  451668*n  +  42120)*a(n-2)  -  161051*(n-5) * (n-4) * (n-3) * (n-2) * (n-1) * (3*n-4) * (3*n-1) * (5*n-14) * (5*n-9) * (5*n-4) * (5*n-3) * (6*n-19) * (6*n-13) * (6*n-7) * (6*n-1)*a(n-6)  -  43923*(n-4) * (n-3) * (n-2) * (n-1) * (2*n-1) * (3*n-7) * (3*n-1) * (5*n-19) * (5*n-9) * (5*n-8) * (5*n-4) * (6*n-25) * (6*n-13) * (6*n-7) * (6*n-1)*a(n-5)  +  5*n*(3*n-7) * (3*n-4) * (5*n-19) * (5*n-14) * (5*n-9) * (5*n-8) * (5*n-4) * (5*n-3) * (5*n-2) * (5*n-1) * (6*n-25) * (6*n-19) * (6*n-13) * (6*n-7)*a(n) = 0. - _Vaclav Kotesovec_, Aug 09 2013

%F a(n) ~ 11^n / sqrt(20*Pi*n). - _Vaclav Kotesovec_, Aug 09 2013

%F a(n) = Sum_{k = 0..floor(n/2)} (-1)^k * binomial(n, k)*binomial(6*n-11*k-1, n-1). - _Peter Bala_, Oct 16 2024

%e Some solutions for n=6

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

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

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

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

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

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

%p seq(add((-1)^k * binomial(n, k)*binomial(6*n-11*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,0,10}]^n],x^(5*n)],{n,1,20}] (* _Vaclav Kotesovec_, Aug 09 2013 *)

%o (PARI) {a(n) = polcoeff((sum(k=0, 10, x^k))^n, 5*n, x)} \\ _Seiichi Manyama_, Dec 14 2018

%Y Column 5 of A201552.

%Y Cf. A001405, A002426, A005190, A005191, A018901, A025012, A025013, A025014, A025015, A225779, A201550.

%K nonn

%O 0,3

%A _R. H. Hardin_, Dec 02 2011

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