A210034 Triangle of coefficients of polynomials v(n,x) jointly generated with A210033; see the Formula section.

1, 2, 1, 4, 2, 1, 7, 5, 2, 1, 12, 10, 6, 2, 1, 20, 20, 13, 7, 2, 1, 33, 38, 29, 16, 8, 2, 1, 54, 71, 60, 39, 19, 9, 2, 1, 88, 130, 122, 86, 50, 22, 10, 2, 1, 143, 235, 241, 187, 116, 62, 25, 11, 2, 1, 232, 420, 468, 392, 267, 150, 75, 28, 12, 2, 1, 376, 744, 894, 806

Offset: 1

Author

Clark Kimberling, Mar 16 2012

Keywords

• nonn
• tabl

Comments

For a discussion and guide to related arrays, see A208510.

From Gus Wiseman, Jun 29 2025: (Start)

This appears to be the number of subsets of {1..n} with k>0 maximal anti-runs (sequences of consecutive elements increasing by more than 1). For example, the subset {1,2,4,5} has maximal anti-runs ((1),(2,4),(5)) so is counted under T(5,3). Row n = 5 counts the following:

{1} {1,2} {1,2,3} {1,2,3,4} {1,2,3,4,5}

{2} {2,3} {2,3,4} {2,3,4,5}

{3} {3,4} {3,4,5}

{4} {4,5} {1,2,3,5}

{5} {1,2,4} {1,2,4,5}

{1,3} {1,2,5} {1,3,4,5}

{1,4} {1,3,4}

{1,5} {1,4,5}

{2,4} {2,3,5}

{2,5} {2,4,5}

{3,5}

{1,3,5}

For runs instead of anti-runs we have A034839, with n A202064. For reversed partitions instead of subsets we have A268193. (End)

Examples

			First five rows:
  1
  2    1
  4    2    1
  7    5    2   1
  12   10   6   2   1
First three polynomials v(n,x): 1, 2 + x, 4 + 2*x + x^2.
		

Crossrefs

Cf. A210033, A208510.

Column k = 1 is A000071.

Row sums are A000225.

Column k = 2 is A001629.

Column k = 3 is A055243.

The version including k = 0 is A384893.

A034839 counts subsets by number of maximal runs, see also A202023, A202064.

A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.

A384877 gives lengths of maximal anti-runs of binary indices, firsts A384878.

Cf. A053538, A116674, A119900, A268193, A384177, A384879, A384889, A384905.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210033 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210034 *)

Formula

u(n,x)=u(n-1,x)+v(n-1,x)+1,

v(n,x)=u(n-1,x)+x*v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.