A202064 Triangle T(n,k), read by rows, given by (2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 2, 0, 3, 1, 0, 4, 4, 0, 0, 5, 10, 1, 0, 0, 6, 20, 6, 0, 0, 0, 7, 35, 21, 1, 0, 0, 0, 8, 56, 56, 8, 0, 0, 0, 0, 9, 84, 126, 36, 1, 0, 0, 0, 0, 10, 120, 252, 120, 10, 0, 0, 0, 0, 0, 11, 165, 462, 330, 55, 1, 0, 0, 0, 0, 0

Offset: 0

Author

Philippe Deléham, Dec 10 2011

Keywords

• nonn
• tabl

Comments

Riordan array (x/(1-x)^2, x^2/(1-x)^2).

Mirror image of triangle in A119900.

A203322*A130595 as infinite lower triangular matrices. - Philippe Deléham, Jan 05 2011

From Gus Wiseman, Jul 07 2025: (Start)

Also the number of subsets of {1..n} containing n with k maximal runs (sequences of consecutive elements increasing by 1). For example, row n = 5 counts the following subsets:

{5} {1,5} {1,3,5}

{4,5} {2,5}

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

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

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

{2,3,5}

{2,4,5}

{1,2,3,5}

{1,2,4,5}

{1,3,4,5}

For anti-runs instead of runs we have A053538.

Without requiring n see A210039, A202023, reverse A098158, A109446.

(End)

Examples

			Triangle begins :
1
2, 0
3, 1, 0
4, 4, 0, 0
5, 10, 1, 0, 0
6, 20, 6, 0, 0, 0
7, 35, 21, 1, 0, 0, 0
8, 56, 56, 8, 0, 0, 0, 0
		

Crossrefs

Cf. A007318, A005314 (antidiagonal sums), A119900, A084938, A130595, A203322.

Column k = 1 is A000027.

Row sums are A000079.

Column k = 2 is A000292.

Without zeros we have A034867.

Last nonzero term in each row appears to be A124625.

A034839 counts subsets by number of maximal runs, for anti-runs A384893.

A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.

Cf. A000045, A000071, A001629, A010027, A053538, A208342, A210034, A245563, A268193, A384177, A384890.

Cf. A098158, A109446, A202023, A210039.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Length[Split[#,#2==#1+1&]]==k&]],{n,12},{k,n}] (* Gus Wiseman, Jul 07 2025 *)

Formula

G.f.: 1/((1-x)^2-y*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = A000027(n+1), A000079(n), A000129(n+1), A002605(n+1), A015518(n+1), A063727(n), A002532(n+1), A083099(n+1), A015519(n+1), A003683(n+1), A002534(n+1), A083102(n), A015520(n+1), A091914(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 10, 11, 12, 13 respectively.

T(n,k) = binomial(n+1,2k+1).

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 15 2012