A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Author

Philippe Deléham, Dec 10 2011

Keywords

• nonn
• tabl

Comments

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

A skewed version of triangular array A085478.

Mirror image of triangle in A098158.

Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.

From Gus Wiseman, Jul 08 2025: (Start)

After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:

{} {1} {1,3} . . .

{2} {1,4}

{3} {2,4}

{4} {1,2,4}

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

{2,3}

{3,4}

{1,2,3}

{2,3,4}

{1,2,3,4}

Requiring n-1 gives A202064.

For anti-runs instead of runs we have A384893.

(End)

Examples

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0
		

Crossrefs

Column k = 1 is A000217.

Column k = 2 is A000332.

Row sums are A011782 (or A000079 shifted right).

Removing all zeros gives A034839 (requiring n-1 A034867).

Last nonzero term in each row appears to be A093178, requiring n-1 A124625.

Reversing rows gives A098158, without zeros A109446.

Without the k = 0 column we get A210039.

Row maxima appear to be A214282.

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

A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.

Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

Formula

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

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

T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if n

T(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

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