A318274 Triangle read by rows: T(n,k) = n for 0 < k < n and T(n,0) = T(n,n) = 1.

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 4, 4, 1, 1, 5, 5, 5, 5, 1, 1, 6, 6, 6, 6, 6, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 8, 8, 8, 8, 8, 8, 8, 1, 1, 9, 9, 9, 9, 9, 9, 9, 9, 1, 1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 1, 12

Offset: 0

Author

Franck Maminirina Ramaharo, Aug 23 2018

Keywords

• nonn
• easy
• tabl

Comments

T(n,k) is the number of binary bitonic words of length n having k letters 1.

Draw a circular rosette such that all the circles contain the rosette's center. Then T(n,k) is also the number of regions in the plane located inside k circles. In fact, a region can be encoded by a binary bitonic word as follows: label each circle from 1 to n in clockwise or counterclockwise order, then write a length n binary word such that the i-th letter indicates whether the concerned region does (write 1) or does not (write 0) lie inside the i-th circle.

Row n is a partition of A014206(n-1) for n > 0.

Examples

			Triangle begins:
n\k| 0  1  2  3  4  5  6  7  8
---+--------------------------
0  | 1
1  | 1  1
2  | 1  2  1
3  | 1  3  3  1
4  | 1  4  4  4  1
5  | 1  5  5  5  5  1
6  | 1  6  6  6  6  6  1
7  | 1  7  7  7  7  7  7  1
8  | 1  8  8  8  8  8  8  8  1
...
For n = 5, the binary bitonic words are
(k = 0) 00000;
(k = 1) 10000, 01000, 00100, 00010, 00001;
(k = 2) 11000, 01100, 00110, 00011, 10001;
(k = 3) 11100, 01110, 00111, 10011, 11001;
(k = 4) 11110, 01111, 10111, 11011, 11101;
(k = 5) 11111.
		

Links

• N. Alon, H. Last, R. Pinchasi and M. Sharir, On the complexity of arrangements of circles in the plane, Discrete Comput. Geom. Vol. 26 (2001), 465-492.
• K. E. Batcher, Sorting networks and their applications, Proceed. AFIPS Spring Joint Comput. Conf. 32 (1968), 307-314.
• W. Denton, Intersecting circles.
• D. Kinsela, Plane division by Lines AND Circles (Problem, Analysis and Solution).
• H. W. Lang, Bitonic sequences.
• F. Ramaharo, Enumerating the states of the twist knot, arXiv:1712.06543 [math.CO], 2017.
• Franck Maminirina Ramaharo, Illustration of initial terms
• P. Rosin, Rosettes and other arrangements of circles, Nexus Network Journal Vol. 3 (2001), 113-126.
• Eric Weisstein's World of Mathematics, Plane Division by Circles.

Crossrefs

Row sums: A014206 preceded by 1.

Cf. A007318, A128227.

Programs

Mathematica

Table[If[k == n || k == 0, 1, n], {n, 0, 20}, {k, 0, n}] // Flatten

Maxima

T(n, k) := if k = 0 or k = n then 1 else if k < n then n else 0$
for n:0 thru 10 do print(makelist(T(n, k), k, 0, n));

PARI

T(n,k) = if ((k==0) || (k==n), 1, n);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Aug 25 2018

Python

from math import isqrt
def A318274(n): return 1 if 0<=(k:=n+1<<1)-(r:=(m:=isqrt(k))*(m+1))<=2 else m-(k<=r) # Chai Wah Wu, Nov 09 2024

Formula

The n-th row are the coefficients in the expansion of 1 + x^n + n*x*(1 - x^(n - 1))/(1 - x), n > 0.

G.f. for column k > 0: (((1 - k)*x^2 - (1 - k)*x + 1)*x^k)/(x - 1)^2.

T(n+1,n-k) - n + k = A128227(n,k).