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A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.

%I A167613 #12 Oct 20 2023 06:46:43
%S A167613 0,0,0,1,1,1,0,-1,-2,-3,0,0,1,3,6,-1,-1,-1,-2,-5,-11,0,1,2,3,5,10,21,
%T A167613 0,0,-1,-3,-6,-11,-21,-42,1,1,1,2,5,11,22,43,85,0,-1,-2,-3,-5,-10,-21,
%U A167613 -43,-86,-171,0,0,1,3,6,11,21,42,85,171,342,-1,-1,-1,-2,-5,-11,-22,-43,-85,-170,-341,-683,0,1,2,3,5,10,21,43,86,171,341,682,1365
%N A167613 Array T(n,k) read by antidiagonals: the k-th term of the n-th difference of A131531.
%C A167613 The array contains A131708(0) in diagonal 0, then -A024495(0..1) in diagonal 1, then A024493(0..2) in diagonal 2, then -A131708(0..3), then A024495(0..4), then -A024493(0..5).
%F A167613 T(0,k) = A131531(k). T(n,k) = T(n-1,k+1) - T(n-1,k), n > 0.
%F A167613 T(n,n) = A001045(n). T(n,n+1) = -A001045(n). T(n,n+2) = A078008(n).
%F A167613 T(n,0) = -T(n,3) = (-1)^(n+1)*A024495(n).
%F A167613 T(n,1) = (-1)^(n+1)*A131708(n).
%F A167613 T(n,2) = (-1)^n*A024493(n).
%F A167613 T(n,k+6) = T(n,k).
%F A167613 a(n) = A131708(0), -A024495(0,1), A024493(0,1,2), -A131708(0,1,2,3), A024495(0,1,2,3,4), -A024493(0,1,2,3,4,5).
%e A167613 The table starts in row n=0 with columns k >= 0 as:
%e A167613 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0, -1, 0, 0, 1, 0, 0 A131531
%e A167613 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, 0, 1, -1 A092220
%e A167613 1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2, 1, -1, 2, -1, 1, -2 A131556
%e A167613 -3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3, 3, -2, 3, -3, 2, -3 A164359
%e A167613 6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5, 5, -6, 5, -5, 6, -5
%e A167613 -11, 10, -11, 11, -10, 11, -11, 10, -11, 11, -10, 11, -11, 10, -11
%e A167613 21, -21, 22, -21, 21, -22, 21, -21, 22, -21, 21, -22, 21, -21, 22
%p A167613 A131531 := proc(n) op((n mod 6)+1,[0,0,1,0,0,-1]) ; end proc:
%p A167613 A167613 := proc(n,k) option remember; if n= 0 then A131531(k); else procname(n-1,k+1)-procname(n-1,k) ; end if;end proc: # _R. J. Mathar_, Dec 17 2010
%t A167613 nmax = 13;
%t A167613 A131531 = Table[{0, 0, 1, 0, 0, -1}, {nmax}] // Flatten;
%t A167613 T[n_] := T[n] = Differences[A131531, n];
%t A167613 T[n_, k_] := T[n][[k]];
%t A167613 Table[T[n-k, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Oct 20 2023 *)
%Y A167613 Cf. A167617 (antidiagonal sums).
%K A167613 tabl,easy,sign
%O A167613 0,9
%A A167613 _Paul Curtz_, Nov 07 2009