A026519 Irregular triangular array T read by rows: T(n, k) = T(n-1, k-2) + T(n-1, k) if (n mod 2) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), with T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor((n+1)/2).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 4, 4, 2, 1, 1, 2, 5, 6, 8, 6, 5, 2, 1, 1, 3, 8, 13, 19, 20, 19, 13, 8, 3, 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, 4, 14, 32, 65, 104, 150, 180, 196, 180, 150, 104, 65, 32, 14, 4, 1

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• tabf

Comments

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

Examples

			First 5 rows:
1
1 ... 1 ... 1
1 ... 1 ... 2 ... 1 ... 1
1 ... 2 ... 4 ... 4 ... 4 ... 2 ... 1
1 ... 2 ... 5 ... 6 ... 8 ... 6 ... 5 ... 2 ... 1
		

Links

• Clark Kimberling, Table of n, a(n) for n = 0..10200 (Rows 0..100, flattened) [Offset changed to 0 by _Georg Fischer_, Mar 01 2022]
• Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
• Veronika Irvine, Stephen Melczer, and Frank Ruskey, Vertically constrained Motzkin-like paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A026552, A026536, A026568, A026584, A027926.

Cf. A026520, A026521, A026522, A026523, A026524, A026525, A026526, A026527, A026528, A026529, A026530, A026531, A026533, A026534, A027262, A027263, A027264, A027265, A027266.

Programs

Mathematica

z = 12; t[n_, 0]:= 1; t[n_, k_]:= 1/; k==2n; t[n_, 1]:= Floor[(n+1)/2]; t[n_, k_] := Floor[(n+1)/2] /; k==2n-1; t[n_, k_]:= t[n, k]= If[EvenQ[n], t[n-1, k-2] + t[n-1, k], t[n-1, k-2] + t[n-1, k-1] + t[n-1, k]];
u = Table[t[n, k], {n, 0, z}, {k, 0, 2n}];
TableForm[u]  (* A026519 array *)
Flatten[u] (* A026519 sequence *)

Sage

@CachedFunction
def T(n,k): # T = A026552
    if (k==0 or k==2*n): return 1
    elif (k==1 or k==2*n-1): return (n+1)//2
    elif (n%2==0): return T(n-1, k) + T(n-1, k-2)
    else: return T(n-1, k) + T(n-1, k-1) + T(n-1, k-2)
flatten([[T(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, Dec 19 2021

Formula

T(n, k) = T(n-1, k-2) + T(n-1, k) if (n mod 2) = 0, otherwise T(n-1, k-2) + T(n-1, k-1) + T(n-1, k), with T(n, 0) = T(n, 2*n) = 1, T(n, 1) = T(n, 2*n-1) = floor((n+1)/2).

Extensions

Updated by Clark Kimberling, Aug 29 2014

Offset changed to 0 by G. C. Greubel, Dec 19 2021