Template:Tiara pair Amethyst/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 1 1 0 2 2 1 1 0 1 1 4 3 1 1 0 7 14 7 0 1 1 32 4 1 1 0 35 140 273 448 715 870 715 448 273 140 35 0 1 1 4096 5 1 1 0 155 1240 6293 27776 105183 330460 876525 2011776 4032015 7063784 10855425 14721280 17678835 18796230 17678835 14721280 10855425 7063784 4032015 2011776 876525 330460 105183 27776 6293 1240 155 0 1 1 134217728

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 0 2 2 1 1 0 1 1 4 3 1 1 0 7 14 7 0 1 1 32 4 1 1 0 35 140 273 448 715 870 715 448 273 140 35 0 1 1 4096 5 1 1 0 155 1240 6293 27776 105183 330460 876525 2011776 4032015 7063784 10855425 14721280 17678835 18796230 17678835 14721280 10855425 7063784 4032015 2011776 876525 330460 105183 27776 6293 1240 155 0 1 1 134217728

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 0 1 1 2 1 0 1 2 3 1 14 0 1 16 4 1 870 140 0 1 1012 5 1 18796230 330460 1240 0 1 19127932

plain text

rows: [ 1, 1], [ 1, 1, 0], [ 1, 1, 0, 1, 1], [ 1, 1, 0, 7, 14, 7, 0, 1, 1], [ 1, 1, 0, 35, 140, 273, 448, 715, 870, 715, 448, 273, 140, 35, 0, 1, 1], [ 1, 1, 0, 155, 1240, 6293, 27776, 105183, 330460, 876525, 2011776, 4032015, 7063784, 10855425, 14721280, 17678835, 18796230, 17678835, 14721280, 10855425, 7063784, 4032015, 2011776, 876525, 330460, 105183, 27776, 6293, 1240, 155, 0, 1, 1] central values:   (rational weight = 1/2) [1, 0, 14, 870, 18796230] diagonal:   (integer weight = arity) [ 1, 1, 0, 7, 140, 6293] row sums: [2, 2, 4, 32, 4096, 134217728] unit fractions: [1], [0, 1], [1, 0, 1], [1, 14, 0, 1], [1, 870, 140, 0, 1], [1, 18796230, 330460, 1240, 0, 1] row sums of unit fractions: [1, 1, 2, 16, 1012, 19127932]

content above, documentation below

A340312 with row sums A300361 = $2^{2^{n}-n}$

The central values are similar to those of A054724. See {{Tiara pair Emerald/wave}}.