A026268 Triangle, T(n, k): T(n,k) = 1 for n < 3, T(3,1) = T(3,2) = T(3,3) = 2, T(n,0) = 1, T(n,1) = n-1, T(n,n) = T(n-1,n-2) + T(n-1,n-1), otherwise T(n,k) = T(n-1,k-2) + T(n-1,k-1) + T(n-1,k), read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 4, 1, 4, 9, 14, 15, 10, 1, 5, 14, 27, 38, 39, 25, 1, 6, 20, 46, 79, 104, 102, 64, 1, 7, 27, 72, 145, 229, 285, 270, 166, 1, 8, 35, 106, 244, 446, 659, 784, 721, 436, 1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157, 1, 10, 54, 202, 578, 1330, 2530, 4034, 5402, 5994, 5262, 3098

Offset

0,8

Comments

a(n) = number of strings s(0)..s(n) such that s(n) = n-k, where s(0) = 0, s(1) = 1, |s(i)-s(i-1)| <= 1 for i >= 2; |s(2)-s(1)| = 1, and |s(3)-s(2)| = 1 if s(2) = 1.

Links

Clark Kimberling, Rows 0..100, flattened

Index entries for triangles and arrays related to Pascal's triangle

Formula

From G. C. Greubel, Sep 24 2022: (Start)

T(n, 1) = A000027(n-1), n >= 1.

T(n, 2) = A212342(n-1), n >= 2.

T(n, n-1) = A026270(n), n >= 2.

T(n, n-2) = A026288(n), n >= 2.

T(n, n-3) = A026289(n), n >= 3.

T(n, n-4) = A026290(n), n >= 4.

T(n, n) = A026269(n), n >= 2.

T(n, floor(n/2)) = A026297(n), n >= 0.

T(2*n, n) = A026292(n).

T(2*n, n-1) = A026295(n), n >= 1.

T(2*n, n+1) = A026296(n), n >= 1.

T(2*n-1, n-1) = A026291(n), n >= 2.

T(3*n, n) = A026293(n), n >= 0.

T(4*n, n) = A026294(n), n >= 0.

Sum_{k=0..n} T(n, k) = A026299(n-1), n >= 3.(End)

Example

Triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,   2;
  1, 3,  5,   6,   4;
  1, 4,  9,  14,  15,  10;
  1, 5, 14,  27,  38,  39,   25;
  1, 6, 20,  46,  79, 104,  102,   64;
  1, 7, 27,  72, 145, 229,  285,  270,  166;
  1, 8, 35, 106, 244, 446,  659,  784,  721,  436;
  1, 9, 44, 149, 385, 796, 1349, 1889, 2164, 1941, 1157;

Mathematica

T[n_, k_]:= T[n, k]= If[n<3 || k==0, 1, If[k==1, n-1, If[k==2, (n^2-n-2)/2 + Boole[n==2], If[k==n, T[n-1, n-2] +T[n-1, n-1], T[n-1, k-2] + T[n-1, k-1] + T[n -1, k] ]]]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}]//Flatten (* corrected by G. C. Greubel, Sep 24 2022 *)

Prog

(Magma)
f:= func< n | n eq 2 select 1 else (n^2 -n -2)/2 >;
function T(n, k) // T = A026268
  if k eq 0 or n lt 3 then return 1;
  elif k eq 1 then return n-1;
  elif k eq 2 then return f(n);
  elif k eq n then return T(n-1, n-2) + T(n-1, n-1);
  else return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k);
  end if; return T;
end function;
[T(n, k): k in [0..n], n in [0..14]]; // G. C. Greubel, Sep 24 2022
(SageMath)
def T(n, k): # T = A026268
    if n<3 or k==0: return 1
    elif k==1: return n-1
    elif k==2: return (n^2 -n -2)//2 + int(n==2)
    elif k==n: return T(n-1, n-2) + T(n-1, n-1)
    else: return T(n-1, k-2) + T(n-1, k-1) + T(n-1, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Sep 24 2022

Crossrefs

Cf. A026269, A026270, A026288, A026289, A026290, A026291, A026292, A026293.

Cf. A026294, A026295, A026296, A026297, A026299, A026519, A026536, A026552.

Cf. A026584, A027926, A212342.

Sequence in context: A055253 A103626 A238224 * A089258 A004065 A127496

Adjacent sequences: A026265 A026266 A026267 * A026269 A026270 A026271

Keyword

nonn,tabl

Author

Clark Kimberling

Extensions

Updated by Clark Kimberling, Aug 29 2014

Indices of b-file corrected by Sidney Cadot, Jan 06 2023.

Status

approved