A026082 Irregular triangular array T read by rows: T(n,k) = C(n,k) for k=0..n for n = 0,1,2,3. For n >= 4, T(n,0) = T(n,2n)=1, T(n,1) = T(n,2n-1) = n - 3, T(4,2) = 4, T(4,3) = 3, T(4,4) = 6; T(4,5) = 3, T(4,6)=4; for n >= 5, T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k) for k=2..2n-2.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 1, 4, 3, 6, 3, 4, 1, 1, 1, 2, 6, 8, 13, 12, 13, 8, 6, 2, 1, 1, 3, 9, 16, 27, 33, 38, 33, 27, 16, 9, 3, 1, 1, 4, 13, 28, 52, 76, 98, 104, 98, 76, 52, 28, 13, 4, 1, 1, 5, 18, 45, 93, 156, 226, 278, 300, 278, 226, 156, 93, 45, 18, 5, 1, 1, 6, 24, 68, 156, 294, 475, 660, 804

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabf

Comments

For n >= 4, T(n,k) = number of strings s(0)..s(n) such that s(n) = n - k, s(0) = 0, |s(i)-s(i-1)| = 1 for i=1,2,3 and |s(i)-s(i-1)| <= 1 for i >= 4.

Examples

			First 6 rows:
  1
  1  1
  1  2  1
  1  3  3  1
  1  1  4  3  6  3  4  1  1
  1  2  6  8 12 12 13  8  6  2  1
		

Links

• Clark Kimberling, Rows 0..100, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

First differences of A024996.

Programs

Maple

A026082 := proc(n,k)
    option remember;
    if n < 0 or k < 0 or k > 2*n then
        0 ;
    elif n <= 3 then
        binomial(n,k) ;
    elif n = 4 then
        op(k+1,[1,1,4,3,6,3,4,1,1]) ;
    elif k =0 or k=2*n then
        1 ;
    else
        procname(n-1,k-2)+procname(n-1,k-1)+procname(n-1,k) ;
    end if;
end proc: # R. J. Mathar, Jun 23 2013

Mathematica

z = 15; t[n_, 0] := 1 /; n >= 4; t[n_, 1] := n - 3 /; n >= 4;
t[4, 2] = 4; t[4, 3] = 3; t[4, 4] = 6; t[4, 5] = 3; t[4, 6] = 4;
t[n_, k_] := t[n, k] = Which[0 <= k <= n && 0 <= n <= 3, Binomial[n, k], n
>= 4 && k == 2 n, 1, k == 2 n - 1, n - 3, 2 <= k <= 2 n - 2, t[n - 1, k -
2] + t[n - 1, k - 1] + t[n - 1, k]]; s = Table[Binomial[n, k], {n, 0, 3},
{k, 0, n}]; u = Join[s, Table[t[n, k], {n, 4, z}, {k, 0, 2 n}]];
TableForm[u] (* A026082 array *)
Flatten[u]   (* A026082 sequence *)

Formula

G.f.: (1-y*z)^3 / (1-z*(1+y+y^2)).

Extensions

Updated by Clark Kimberling, Aug 28 2014