Template:Tiara pair Onyx/wave

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 0 1 1 2 2 0 1 4 4 1 10 3 0 1 13 44 67 56 28 8 1 218 4 0 1 40 360 1546 4144 7896 11408 12866 11440 8008 4368 1820 560 120 16 1 64594 5 0 1 121 2680 27550 180096 866432 3308736 10453960 27991600 64472200 129002640 225783740 347370800 471435000 565722640 601080385 565722720 471435600 347373600 225792840 129024480 64512240 28048800 10518300 3365856 906192 201376 35960 4960 496 32 1 4294642034

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 0 1 1 2 2 0 1 4 4 1 10 3 0 1 13 44 67 56 28 8 1 218 4 0 1 40 360 1546 4144 7896 11408 12866 11440 8008 4368 1820 560 120 16 1 64594 5 0 1 121 2680 27550 180096 866432 3308736 10453960 27991600 64472200 129002640 225783740 347370800 471435000 565722640 601080385 565722720 471435600 347373600 225792840 129024480 64512240 28048800 10518300 3365856 906192 201376 35960 4960 496 32 1 4294642034

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 4 1 6 3 1 67 13 1 82 4 1 12866 1546 40 1 14454 5 1 601080385 10453960 27550 121 1 611562018

plain text

rows: [ 1, 1], [ 0, 1, 1], [ 0, 1, 4, 4, 1], [ 0, 1, 13, 44, 67, 56, 28, 8, 1], [ 0, 1, 40, 360, 1546, 4144, 7896, 11408, 12866, 11440, 8008, 4368, 1820, 560, 120, 16, 1], [ 0, 1, 121, 2680, 27550, 180096, 866432, 3308736, 10453960, 27991600, 64472200, 129002640, 225783740, 347370800, 471435000, 565722640, 601080385, 565722720, 471435600, 347373600, 225792840, 129024480, 64512240, 28048800, 10518300, 3365856, 906192, 201376, 35960, 4960, 496, 32, 1] central values:   (rational weight = 1/2) [1, 4, 67, 12866, 601080385] diagonal:   (integer weight = arity) [ 1, 1, 4, 44, 1546, 180096] row sums: [2, 2, 10, 218, 64594, 4294642034] unit fractions: [1], [1, 1], [1, 4, 1], [1, 67, 13, 1], [1, 12866, 1546, 40, 1], [1, 601080385, 10453960, 27550, 121, 1] row sums of unit fractions: [1, 2, 6, 82, 14454, 611562018]

content above, documentation below

🌊    triangle A163353    row sums Lotus (A000371)

Dense (non-degenerate) Boolean functions by weight.

rational weight:

Let C be the central column. Then C(a) + a = 2 · A069954(a−1) = A000984(2^a−1) for a > 0.

integer weight:

Diagonal is A134174.

Column 2 is A003462.   a(n) = (3ⁿ − 1) / 2.

merged weights:

The diagonal 1, 5, 93, 26333... is half of A037267(1...) = 2, 10, 186, 52666...