A026386 Triangular array T read by rows: T(n,0) = T(n,n) = 1 for all n >= 0; T(n,k) = T(n-1,k-1) + T(n-1,k) for even n and k = 1..n-1; T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-1) for odd n and k = 1 ..n-1.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 5, 8, 5, 1, 1, 7, 17, 17, 7, 1, 1, 8, 24, 34, 24, 8, 1, 1, 10, 39, 75, 75, 39, 10, 1, 1, 11, 49, 114, 150, 114, 49, 11, 1, 1, 13, 70, 202, 339, 339, 202, 70, 13, 1, 1, 14, 83, 272, 541, 678, 541, 272, 83, 14, 1, 1, 16

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabl
• easy

Comments

T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = n - 2k, where, for u = 1..n, s(i) is odd if i is odd and |s(i)-s(i-1)| <=1.

Examples

			Rows n=0 through n=7:
  1
  1 ... 1
  1 ... 2 ... 1
  1 ... 4 ... 4 ... 1
  1 ... 5 ... 8 ... 5 ... 1
  1 ... 7 ... 17 .. 17 .. 7 ... 1
  1 ... 8 ... 24 .. 34 .. 24 .. 8 ... 1
  1 ... 10 .. 39 .. 75 .. 75 .. 39 .. 10 ... 1
		

Links

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

Crossrefs

Cf. A007318.

Programs

Maple

A026386 := proc(n,k)
    option remember;
    if k=0 or k = n then
        1;
    elif k <0 or k > n then
        0 ;
    elif type(n,'even') then
        procname(n-1,k-1)+procname(n-1,k) ;
    else
        procname(n-1,k-1)+procname(n-1,k)+procname(n-2,k-1) ;
    end if;
end proc: # R. J. Mathar, Feb 10 2015

Mathematica

z = 12; t[n_, 0] := 1; t[n_, n_] := 1; t[n_, k_] := t[n, k] =
Which[EvenQ[n], t[n - 1, k - 1] + t[n - 1, k], OddQ[n], t[n - 1, k - 1] +
t[n - 1, k] + t[n - 2, k - 1]]; u = Table[t[n, k], {n, 0, z}, {k, 0, n}];
TableForm[u] (* A026386 array *)
Flatten[u]   (* A026386 sequence *)

PARI

T(n)={[Vecrev(p) | p<-Vec((1 + (1 + y)*x - y*x^2)/(1 - (1 + 3*y + y^2)*x^2) + O(x*x^n))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Dec 27 2024

Formula

G.f.: (1 + (1 + y)*x - y*x^2)/(1 - (1 + 3*y + y^2)*x^2). - Andrew Howroyd, Dec 27 2024

Extensions

Updated by Clark Kimberling, Aug 28 2014

Offset corrected by R. J. Mathar, Feb 10 2015