

A026519


Irregular triangular array T read by rows: T(i,0) = T(i,2i)=1 for i >= 0; T(i,1) = T(i,2i1) = floor[(i+1)/2] for i >= 1; for even n >= 2, T(i,j) = T(i1,j2) + T(i1,j) for j = 2..2i2; for odd n >= 3, T(i,j) = T(i1,j2) + T(i1,j1) + T(i1,j) for j = 2..2i2.


26



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
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OFFSET

1,7


COMMENTS

T(n, k) = number of integer strings s(0)..s(n) such that s(0) = 0, s(n) = nk, s(i)s(i1) = 1 if i is even, s(i)s(i1) <= 1 if i is odd.


LINKS

Clark Kimberling, Rows 0..100, flattened
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, Frank Ruskey, Vertically constrained Motzkinlike paths inspired by bobbin lace, arXiv:1804.08725 [math.CO], 2018.
Index entries for triangles and arrays related to Pascal's triangle


EXAMPLE

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


MATHEMATICA

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


CROSSREFS

Cf. A026520, A026521, A026522, A026523, A026524, A026527, A026552, A026536, A026568, A026584, A027926.
Sequence in context: A267383 A272896 A188919 * A025177 A026148 A117211
Adjacent sequences: A026516 A026517 A026518 * A026520 A026521 A026522


KEYWORD

nonn,tabf


AUTHOR

Clark Kimberling


EXTENSIONS

Updated by Clark Kimberling, Aug 29 2014


STATUS

approved



