A201549 Number of arrays of n integers in -5..5 with sum zero.

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

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

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

Crossrefs

Column 5 of A201552.

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

Sequence in context: A126532 A226868 A199678 * A117611 A294362 A363669

Adjacent sequences: A201546 A201547 A201548 * A201550 A201551 A201552

Keyword

nonn

Author

R. H. Hardin, Dec 02 2011

Extensions

a(0)=1 prepended by Seiichi Manyama, Dec 14 2018

Status

approved