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A201549
Number of arrays of n integers in -5..5 with sum zero.
9
1, 1, 11, 91, 891, 8801, 88913, 908755, 9377467, 97464799, 1018872811, 10701243741, 112835748609, 1193692544825, 12663809507129, 134678108144591, 1435345208419771, 15326122342137035, 163920458145421109
OFFSET
0,3
COMMENTS
Also largest coefficient of (1+x+...+x^10)^n. - Vaclav Kotesovec, Aug 09 2013
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..500 (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]
FORMULA
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
a(n) ~ 11^n / sqrt(20*Pi*n). - Vaclav Kotesovec, Aug 09 2013
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
EXAMPLE
Some solutions for n=6
.-5...-5...-1...-2...-1....3....4....1....1...-1....3....4....5....0...-5....5
.-2....4...-1....1....0...-4....2....1....1...-5...-4...-4....1...-3....5...-5
..0...-3....1....3...-1...-4....0...-1....2...-4...-4...-4...-5...-3...-2....5
..4....3....3...-3....4....5....1....2....2....5....3....5...-3....2....2...-5
..5...-4...-1...-4...-4....1...-2...-4...-5....2....0....4....1....4...-4....3
.-2....5...-1....5....2...-1...-5....1...-1....3....2...-5....1....0....4...-3
MAPLE
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
MATHEMATICA
Table[Coefficient[Expand[Sum[x^j, {j, 0, 10}]^n], x^(5*n)], {n, 1, 20}] (* Vaclav Kotesovec, Aug 09 2013 *)
PROG
(PARI) {a(n) = polcoeff((sum(k=0, 10, x^k))^n, 5*n, x)} \\ Seiichi Manyama, Dec 14 2018
KEYWORD
nonn
AUTHOR
R. H. Hardin, Dec 02 2011
EXTENSIONS
a(0)=1 prepended by Seiichi Manyama, Dec 14 2018
STATUS
approved