A193648 T(n,k)=Number of arrays of -k..k integers x(1..n) with every x(i) being in a substring of length 1 or 2 with sum zero. Array listed by antidiagonals.

1, 1, 3, 1, 5, 7, 1, 7, 13, 15, 1, 9, 19, 37, 33, 1, 11, 25, 67, 105, 73, 1, 13, 31, 105, 217, 297, 161, 1, 15, 37, 151, 369, 721, 841, 355, 1, 17, 43, 205, 561, 1393, 2377, 2381, 783, 1, 19, 49, 267, 793, 2361, 5105, 7855, 6741, 1727, 1, 21, 55, 337, 1065, 3673, 9361

Offset

1,3

Links

R. H. Hardin, Table of n, a(n) for n = 1..9999

Formula

Empirical for column k: T(n,k)=2*T(n-1,k)+2*(k-1)*T(n-2,k)+T(n-3,k); with T(1,k)=1, T(2,k)=2*k+1, T(3,k)=6*k+1.

From Robert Israel, May 26 2016: (Start)

G.f. for column k: (x+(2k-1)x^2+x^3)/(1-2x+2(1-k)x^2-x^3).

The recursion for column k can be obtained from this.

G.f. for array: A(x,y) = y/(y-1) - (1-x+x^2)*y*LerchPhi(y,1,(-1+2*x+x^3)/(2*x^2))/(2*x^2). (End)

Example

Table starts
....1.....1.....1......1......1.......1.......1.......1.......1........1
....3.....5.....7......9.....11......13......15......17......19.......21
....7....13....19.....25.....31......37......43......49......55.......61
...15....37....67....105....151.....205.....267.....337.....415......501
...33...105...217....369....561.....793....1065....1377....1729.....2121
...73...297...721...1393...2361....3673....5377....7521...10153....13321
..161...841..2377...5105...9361...15481...23801...34657...48385....65321
..355..2381..7855..18937..38171...68485..113191..175985..260947...372541
..783..6741.25939..69897.153591..295453..517371..844689.1306207..1934181
.1727.19085.85675.258521.621911.1291237.2416835.4187825.6835951.10639421
Some solutions for n=7 k=6
.-6....4....1....2...-5...-2....1....5...-5...-1....4...-3...-4...-6....0....0
..6...-4...-1...-2....5....2...-1...-5....5....1...-4....3....4....6....1....5
.-6....4....1...-4...-3....0....3....5....6...-3...-1....0...-5....0...-1...-5
.-4....2...-3....4....3...-4...-3...-5...-6....3....1....2....5....3...-1....5
..4...-2....3...-1...-3....4....4....4....1...-3....1...-2...-5...-3....1....0
.-4....3...-4....1...-6...-3...-4...-4...-1...-2...-1....3...-5....0...-1...-6
..4...-3....4...-1....6....3....0....4....1....2....1...-3....5....0....0....6

Maple

F:= normal @ gfun:-rectoproc({t(n) = 2*t(n-1)+2*(k-1)*t(n-2)+t(n-3), t(1)=1, t(2)=2*k+1, t(3)=6*k+1}, t(n), remember):
seq(seq(eval(F(j), k=m-j), j=1..m-1), m=2..20); # Robert Israel, May 26 2016

Mathematica

nmax = 12;
col[k_] := col[k] = CoefficientList[(x + (2 k - 1) x^2 + x^3)/
   (1 - 2 x + 2 (1 - k) x^2 - x^3) + O[x]^(nmax + 1), x] // Rest;
T[n_, k_] := col[k][[n]];
Table[T[n - k + 1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jul 24 2022 *)

Crossrefs

Cf. A193641 (column 1) to A193647 (column 7).

Sequence in context: A130418 A038871 A209819 * A221881 A201811 A199898

Adjacent sequences: A193645 A193646 A193647 * A193649 A193650 A193651

Keyword

nonn,tabl

Author

R. H. Hardin, Aug 02 2011

Status

approved