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.

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, 1072, 2157, 1408, 365, 36, 1, 0, 1, 6985, 15701, 12445, 4417, 721, 49, 1, 0, 1, 53218, 129214, 116698, 50944, 11452, 1288, 64, 1, 0, 1, 462331, 1191336, 1183216, 597026, 166716, 25956, 2136, 81, 1, 0

Offset: 0

Author

Philippe Deléham, Nov 19 2011

Keywords

• nonn
• tabl

Comments

Row sums : A000142(n) = n!.

Examples

			Triangle begins :
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, 1072, 2157, 1408, 365, 36, 1, 0
1, 6985, 15701, 12445, 4417, 721, 49, 1, 0
1, 53218, 129214, 116698, 50944, 11452, 1288, 64, 1, 0
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Sergey Kitaev, Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.

Crossrefs

Cf. A000290, A014145,

Programs

Mathematica

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}]];
m = 10;
DELTA[LinearRecurrence[{1, 1, -1}, {1, 0, 2}, m], LinearRecurrence[{0, 1}, {0, 1}, m], m] // Flatten (* Jean-François Alcover, Feb 21 2019 *)

Formula

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.

T(k+2,k)=(k+1)^2 = A000290(k+1).

T(n+1,1)= A014145(n).