A055216 Triangle T(n,k) by rows, n >= 0, 0<=k<=n: T(n,k) = Sum_{i=0..n-k} binomial(n-k,i) *Sum_{j=0..k-i} binomial(i,j).

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 3, 1, 1, 5, 10, 8, 3, 1, 1, 6, 15, 17, 9, 3, 1, 1, 7, 21, 31, 23, 9, 3, 1, 1, 8, 28, 51, 50, 26, 9, 3, 1, 1, 9, 36, 78, 96, 66, 27, 9, 3, 1, 1, 10, 45, 113, 168, 147, 76, 27, 9, 3, 1, 1, 11, 55, 157, 274, 294, 192, 80, 27, 9, 3, 1

Offset: 0

Author

Clark Kimberling, May 07 2000

Keywords

• nonn
• tabl

Comments

T(n,k) is the maximal number of different sequences that can be obtained from a ternary sequence of length n by deleting k symbols.

T(i,j) is the number of paths from (0,0) to (i-j,j) using steps (1 unit right) or (1 unit right and 1 unit up) or (1 unit right and 2 units up).

If m >= 1 and n >= 2, then T(m+n-1,m) is the number of strings (s(1),s(2),...,s(n)) of nonnegative integers satisfying s(n)=m and 0<=s(k)-s(k-1)<=2 for k=2,3,...,n.

T(n,k) is the number of 1100-avoiding 0-1 sequences of length n containing k good 1's. A 1 is bad if it is immediately followed by two or more 1's and then a 0; otherwise it is good. In particular, a 1 with no 0's to its right is good. For example, 110101110111 is 1100-avoiding and only the 1 in position 6 is bad and T(4,3) counts 0111, 1011, 1101. - David Callan, Jul 25 2005

The matrix inverse starts:

1;

-1,1;

1,-2,1;

-1,3,-3,1;

1,-4,6,-4,1;

-2,8,-13,11,-5,1;

8,-30,45,-36,18,-6,1;

-36,137,-207,163,-78,27,-7,1;

192,-732,1112,-884,425,-144,38,-8,1;

- R. J. Mathar, Mar 12 2013

Examples

			8=T(5,2) counts these strings: 013, 023, 113, 123, 133, 223, 233, 333.
Rows:
1;
1,1;
1,2,1;
1,3,3,1;
1,4,6,3,1;
...
		

Links

• D. S. Hirschberg, Algorithms for the longest subsequence problem, J. ACM, 24 (1977), 664-675.
• C. Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 1E.
• V. I. Levenshtein, Efficient reconstruction of sequences from their subsequences or supersequences, J. Combin. Theory Ser. A 93 (2001), no. 2, 310-332.

Crossrefs

Row sums: A008937. Central numbers: T(2n, n)=A027914(n) for n >= 0.

Programs

Maple

A055216 := proc(n,k)
    a := 0 ;
    for i from 0 to n-k do
        a := a+binomial(n-k,i)*add(binomial(i,j),j=0..k-i) ;
    end do:
    a ;
end proc: # R. J. Mathar, Mar 13 2013

Mathematica

T[n_, k_] := Sum[Binomial[n - k, i]*Sum[Binomial[i, j], {j, 0, k - i}], {i, 0, n - k}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 28 2019 *)

Formula

T(i, 0)=T(i, i)=1 for i >= 0; T(i, 1)=T(i, i-1)=i for i >= 2; T(i, j)=T(i-1, j)+T(i-2, j-1)+T(i-3, j-2) for 2<=j<=i-2, i >= 3.

Extensions

Better description and references from N. J. A. Sloane, Aug 05 2000