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%I #17 Jul 25 2022 04:03:13
%S 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,
%T 297,161,1,15,37,151,369,721,841,355,1,17,43,205,561,1393,2377,2381,
%U 783,1,19,49,267,793,2361,5105,7855,6741,1727,1,21,55,337,1065,3673,9361
%N 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.
%H R. H. Hardin, <a href="/A193648/b193648.txt">Table of n, a(n) for n = 1..9999</a>
%F 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.
%F From _Robert Israel_, May 26 2016: (Start)
%F G.f. for column k: (x+(2k-1)x^2+x^3)/(1-2x+2(1-k)x^2-x^3).
%F The recursion for column k can be obtained from this.
%F 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)
%e Table starts
%e ....1.....1.....1......1......1.......1.......1.......1.......1........1
%e ....3.....5.....7......9.....11......13......15......17......19.......21
%e ....7....13....19.....25.....31......37......43......49......55.......61
%e ...15....37....67....105....151.....205.....267.....337.....415......501
%e ...33...105...217....369....561.....793....1065....1377....1729.....2121
%e ...73...297...721...1393...2361....3673....5377....7521...10153....13321
%e ..161...841..2377...5105...9361...15481...23801...34657...48385....65321
%e ..355..2381..7855..18937..38171...68485..113191..175985..260947...372541
%e ..783..6741.25939..69897.153591..295453..517371..844689.1306207..1934181
%e .1727.19085.85675.258521.621911.1291237.2416835.4187825.6835951.10639421
%e Some solutions for n=7 k=6
%e .-6....4....1....2...-5...-2....1....5...-5...-1....4...-3...-4...-6....0....0
%e ..6...-4...-1...-2....5....2...-1...-5....5....1...-4....3....4....6....1....5
%e .-6....4....1...-4...-3....0....3....5....6...-3...-1....0...-5....0...-1...-5
%e .-4....2...-3....4....3...-4...-3...-5...-6....3....1....2....5....3...-1....5
%e ..4...-2....3...-1...-3....4....4....4....1...-3....1...-2...-5...-3....1....0
%e .-4....3...-4....1...-6...-3...-4...-4...-1...-2...-1....3...-5....0...-1...-6
%e ..4...-3....4...-1....6....3....0....4....1....2....1...-3....5....0....0....6
%p 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):
%p seq(seq(eval(F(j),k=m-j),j=1..m-1),m=2..20); # _Robert Israel_, May 26 2016
%t nmax = 12;
%t col[k_] := col[k] = CoefficientList[(x + (2 k - 1) x^2 + x^3)/
%t (1 - 2 x + 2 (1 - k) x^2 - x^3) + O[x]^(nmax + 1), x] // Rest;
%t T[n_, k_] := col[k][[n]];
%t Table[T[n - k + 1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Jul 24 2022 *)
%Y Cf. A193641 (column 1) to A193647 (column 7).
%K nonn,tabl
%O 1,3
%A _R. H. Hardin_, Aug 02 2011