Tiara Emerald

Studies of Boolean functions
sequences related to Boolean functions

Tiara Emerald

integer weight

w a	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	sums
0	1	1																																2
1	1	1	1																															3
2	1	1	3	1	1																													7
3	1	1	7	7	14	7	7	1	1																									46
4	1	1	15	35	140	273	553	715	870	715	553	273	140	35	15	1	1																	4336
5	1	1	31	155	1240	6293	28861	105183	330460	876525	2020239	4032015	7063784	10855425	14743445	17678835	18796230	17678835	14743445	10855425	7063784	4032015	2020239	876525	330460	105183	28861	6293	1240	155	31	1	1	134281216

rational weight

w a	$0$	${\frac {1}{32}}$	${\frac {1}{16}}$	${\frac {3}{32}}$	${\frac {1}{8}}$	${\frac {5}{32}}$	${\frac {3}{16}}$	${\frac {7}{32}}$	${\frac {1}{4}}$	${\frac {9}{32}}$	${\frac {5}{16}}$	${\frac {11}{32}}$	${\frac {3}{8}}$	${\frac {13}{32}}$	${\frac {7}{16}}$	${\frac {15}{32}}$	${\frac {1}{2}}$	${\frac {17}{32}}$	${\frac {9}{16}}$	${\frac {19}{32}}$	${\frac {5}{8}}$	${\frac {21}{32}}$	${\frac {11}{16}}$	${\frac {23}{32}}$	${\frac {3}{4}}$	${\frac {25}{32}}$	${\frac {13}{16}}$	${\frac {27}{32}}$	${\frac {7}{8}}$	${\frac {29}{32}}$	${\frac {15}{16}}$	${\frac {31}{32}}$	$1$	sums
0	1																																1	2
1	1																1																1	3
2	1								1								3								1								1	7
3	1				1				7				7				14				7				7				1				1	46
4	1		1		15		35		140		273		553		715		870		715		553		273		140		35		15		1		1	4336
5	1	1	31	155	1240	6293	28861	105183	330460	876525	2020239	4032015	7063784	10855425	14743445	17678835	18796230	17678835	14743445	10855425	7063784	4032015	2020239	876525	330460	105183	28861	6293	1240	155	31	1	1	134281216

unit fractions

w a	$1$	${\frac {1}{2}}$	${\frac {1}{4}}$	${\frac {1}{8}}$	${\frac {1}{16}}$	${\frac {1}{32}}$	sums
0	1						1
1	1	1					2
2	1	3	1				5
3	1	14	7	1			23
4	1	870	140	15	1		1027
5	1	18796230	330460	1240	31	1	19127963

plain text

rows:
[ 1, 1],
[ 1, 1, 1],
[ 1, 1, 3, 1, 1],
[ 1, 1, 7, 7, 14, 7, 7, 1, 1],
[ 1, 1, 15, 35, 140, 273, 553, 715, 870, 715, 553, 273, 140, 35, 15, 1, 1],
[ 1, 1, 31, 155, 1240, 6293, 28861, 105183, 330460, 876525, 2020239, 4032015, 7063784, 10855425, 14743445, 17678835, 18796230, 17678835, 14743445, 10855425, 7063784, 4032015, 2020239, 876525, 330460, 105183, 28861, 6293, 1240, 155, 31, 1, 1]

central values: (rational weight = 1/2)
[1, 3, 14, 870, 18796230]

diagonal: (integer weight = arity)
[ 1, 1, 3, 7, 140, 6293]

row sums:
[2, 3, 7, 46, 4336, 134281216]

unit fractions:
[1],
[1, 1],
[1, 3, 1],
[1, 14, 7, 1],
[1, 870, 140, 15, 1],
[1, 18796230, 330460, 1240, 31, 1]

row sums of unit fractions: [1, 2, 5, 23, 1027, 19127963]

The central column (and thus column 1 of the slab) is sequence Saffron, the number of balanced families.

The row sums are Thyme.

(a, w) ↦ families

Emerald(a, w)
EmeraldDrop(a, w)

is the number of families with

arity
adicity

a and weight w.

Triangle EmeraldSlab

triangle EmeraldSlab
k n	0	1	2	3	4	5	6	7	sums
0	1								1
1	1	1							2
2	1	3	1						5
3	1	14	7	1					23
4	1	870	140	15	1				1027
5	1	18796230	330460	1240	31	1			19127963
6	1	28634752793916486	7633243745820	69194232	10416	63	1		28642386106867019
7	1	187118328452563149209991044344449606	11545366574237052418777217820	729242357526446712	11170182384	85344	127	1	187118339997929724176285831818381995

The diagonals are:

1, 1, 1, ...
1, 3, 7, 15, 31, ... (A000225)
1, 14, 140, 1240, 10416, 85344, ... (A016290)
1, 870, 330460, 69194232, 11170182384