Template:Boolf triangle Magnolia

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

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

(merged weights)
w 0 1 2 3...4 5...8 9...16 17...32 sums k a -1 0 1 2 3 4 5 0 1 1 2 1 0 1 1 2 2 0 1 4 5 10 3 0 1 13 111 93 218 4 0 1 40 1906 36314 26333 64594 5 0 1 121 30230 14809224 2432859005 1846943453 4294642034

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🌊    triangle Magnolia (A163353)    row sums Lotus (A000371)

summation of {{Boolf triangle Myrtle}}

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...