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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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

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

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

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

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Last modified May 8 13:26 EDT 2021. Contains 343666 sequences. (Running on oeis4.)