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A200545 Triangle T(n,k), read by rows, given by (1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.

%I A200545 #17 Feb 21 2019 11:25:47
%S A200545 1,1,0,1,1,0,1,4,1,0,1,13,9,1,0,1,46,56,16,1,0,1,199,334,160,25,1,0,1,
%T A200545 1072,2157,1408,365,36,1,0,1,6985,15701,12445,4417,721,49,1,0,1,53218,
%U A200545 129214,116698,50944,11452,1288,64,1,0,1,462331,1191336,1183216,597026,166716,25956,2136,81,1,0
%N A200545 Triangle T(n,k), read by rows, given by (1,0,2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938.
%C A200545 Row sums : A000142(n) = n!.
%H A200545 Alois P. Heinz, <a href="/A200545/b200545.txt">Rows n = 0..140, flattened</a>
%H A200545 Sergey Kitaev, Philip B. Zhang, <a href="https://arxiv.org/abs/1811.07679">Distributions of mesh patterns of short lengths</a>, arXiv:1811.07679 [math.CO], 2018.
%F A200545 Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A172485(n+1), A146559(n), A000012(n), A000142(n), A003319(n), A111529(n), A111530(n), A111531(n), A111532(n), A111533(n) for x = -2,-1,0,1,2,3,4,5,6,7 respectively.
%F A200545 T(k+2,k)=(k+1)^2 = A000290(k+1).
%F A200545 T(n+1,1)= A014145(n).
%e A200545 Triangle begins :
%e A200545 1
%e A200545 1, 0
%e A200545 1, 1, 0
%e A200545 1, 4, 1, 0
%e A200545 1, 13, 9, 1, 0
%e A200545 1, 46, 56, 16, 1, 0
%e A200545 1, 199, 334, 160, 25, 1, 0
%e A200545 1, 1072, 2157, 1408, 365, 36, 1, 0
%e A200545 1, 6985, 15701, 12445, 4417, 721, 49, 1, 0
%e A200545 1, 53218, 129214, 116698, 50944, 11452, 1288, 64, 1, 0
%t A200545 DELTA[r_, s_, m_] := Module[{p, q, t, x, y}, q[k_] := x*r[[k + 1]] + y*s[[k + 1]]; p[0, _] = 1; p[_, -1] = 0; p[n_ /; n >= 1, k_ /; k >= 0] := p[n, k] = p[n, k - 1] + q[k]*p[n - 1, k + 1] // Expand; t[n_, k_] := Coefficient[p[n, 0], x^(n - k)*y^k]; t[0, 0] = p[0, 0]; Table[t[n, k], {n, 0, m}, {k, 0, n}]];
%t A200545 m = 10;
%t A200545 DELTA[LinearRecurrence[{1, 1, -1}, {1, 0, 2}, m], LinearRecurrence[{0, 1}, {0, 1}, m], m] // Flatten (* _Jean-François Alcover_, Feb 21 2019 *)
%Y A200545 Cf. A000290, A014145,
%K A200545 nonn,tabl
%O A200545 0,8
%A A200545 _Philippe Deléham_, Nov 19 2011