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

0,8

Comments

Row sums : A000142(n) = n!.

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.

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).

Example

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

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 *)

Crossrefs

Cf. A000290, A014145,

Sequence in context: A362868 A055105 A348600 * A294522 A058710 A281891

Adjacent sequences: A200542 A200543 A200544 * A200546 A200547 A200548

Keyword

nonn,tabl

Author

Philippe Deléham, Nov 19 2011

Status

approved