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Bernoulli's triangle

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Bernoulli's triangle is the triangle of partial sums of binomial coefficients, i.e. partial sums across rows of Pascal's triangle (see A007318). For example, in Pascal's triangle, the row for is 1 3 3 1. In Bernoulli's triangle, 1 is 1, 1 + 3 is 4, 1 + 3 + 3 is 7 and 1 + 3 + 3 + 1 is 8, giving 1 4 7 8 as the row for in Bernoulli's triangle.

As the row sums of Pascal's triangle give the powers of two (see A000079), so the rightmost falling diagonal of Bernoulli's triangle contains the powers of two, and the second rightmost falling diagonal contains the Mersenne numbers (see A000225).

B E R N O U L L I ' S T R I A N G L E Row sums

(A001792())

0 1 1
1 1 2 3
2 1 3 4 8
3 1 4 7 8 20
4 1 5 11 15 16 48
5 1 6 16 26 31 32 112
6 1 7 22 42 57 63 64 256
7 1 8 29 64 99 120 127 128 576
8 1 9 37 93 163 219 247 255 256 1280
9 1 10 46 130 256 382 466 502 511 512 2816
10 1 11 56 176 A000127 starting with : Maximal number of regions obtained by joining points around a circle by straight lines. 6144
11 1 12 67 A000125 starting with : Cake numbers: maximal number of pieces resulting from planar cuts through a cube or cake. 13312
12 1 13 A000124 starting with : Central polygonal numbers 28672
13 1 A000027 starting with : Each positive integer greater than 1 61440
14 A000012: Continued fraction for the golden ratio 131072


Bernoulli's triangle recurrence equation

The leftmost entries are set to 1, i.e. the leftmost entries of Pascal's triangle. The rightmost entries are set to , i.e. the row sums of Pascal's triangle. Then a recurrence equation identical to Pascal's triangle is applied.

Bernoulli's triangle formula

and

Bernoulli's triangle rows

Bernoulli's triangle read by rows gives the infinite sequence of finite sequences

{{1}, {1, 2}, {1, 3, 4}, {1, 4, 7, 8}, {1, 5, 11, 15, 16}, {1, 6, 16, 26, 31, 32}, {1, 7, 22, 42, 57, 63, 64}, {1, 8, 29, 64, 99, 120, 127, 128}, {1, 9, 37, 93, 163, 219, 247, 255, 256}, ...}

whose concatenation gives the infinite sequence (see A008949)

{1, 1, 2, 1, 3, 4, 1, 4, 7, 8, 1, 5, 11, 15, 16, 1, 6, 16, 26, 31, 32, 1, 7, 22, 42, 57, 63, 64, 1, 8, 29, 64, 99, 120, 127, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256, 1, 10, 46, 130, 256, 382, 466, 502, 511, 512, ...}

Bernoulli's triangle rows sums

Bernoulli's triangle rows sums gives the infinite sequence (see A001792)

{1, 3, 8, 20, 48, 112, 256, 576, 1280, 2816, 6144, 13312, 28672, 61440, 131072, 278528, 589824, 1245184, 2621440, 5505024, 11534336, 24117248, 50331648, 104857600, 218103808, 452984832, ...}

which is given by the formula

The generating function is

Bernoulli's triangle rows alternating sign sums

Bernoulli's triangle rising diagonals

The table gives the th member, , of the th, , rising diagonal (0 for leftmost).

Bernoulli's triangle rising diagonals sequences
sequences A-number
0 {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...} A000012()
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, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, ...} A000027()
2 {4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, ...} A000124()
3 {8, 15, 26, 42, 64, 93, 130, 176, 232, 299, 378, 470, 576, 697, 834, 988, 1160, 1351, 1562, 1794, 2048, 2325, 2626, 2952, 3304, 3683, 4090, 4526, 4992, 5489, 6018, 6580, 7176, ...} A000125()
4 {16, 31, 57, 99, 163, 256, 386, 562, 794, 1093, 1471, 1941, 2517, 3214, 4048, 5036, 6196, 7547, 9109, 10903, 12951, 15276, 17902, 20854, 24158, 27841, 31931, 36457, 41449, ...} A000127()
5 {32, 63, 120, 219, 382, 638, 1024, 1586, 2380, 3473, 4944, 6885, 9402, 12616, 16664, 21700, 27896, 35443, 44552, 55455, 68406, 83682, 101584, 122438, 146596, 174437, ...} A006261()
6 {64, 127, 247, 466, 848, 1486, 2510, 4096, 6476, 9949, 14893, 21778, 31180, 43796, 60460, 82160, 110056, 145499, 190051, 245506, 313912, 397594, 499178, 621616, 768212, ...} A008859()
7 {128, 255, 502, 968, 1816, 3302, 5812, 9908, 16384, 26333, 41226, 63004, 94184, 137980, 198440, 280600, 390656, 536155, 726206, 971712, 1285624, 1683218, 2182396, ...} A008860()
8 {256, 511, 1013, 1981, 3797, 7099, 12911, 22819, 39203, 65536, 106762, 169766, 263950, 401930, 600370, 880970, 1271626, 1807781, 2533987, 3505699, 4791323, 6474541, ...} A008861()
9 {512, 1023, 2036, 4017, 7814, 14913, 27824, 50643, 89846, 155382, 262144, 431910, 695860, 1097790, 1698160, 2579130, 3850756, 5658537, 8192524, 11698223, 16489546, ...} A008862()
10 {1024, 2047, 4083, 8100, 15914, 30827, 58651, 109294, 199140, 354522, 616666, 1048576, 1744436, 2842226, 4540386, 7119516, 10970272, 16628809, 24821333, 36519556, ...} A008863()


Bernoulli's triangle rising diagonals formulae and values
Formulae


Generating

function


Differences

Partial sums

Partial sums of reciprocals

Sum of Reciprocals

0
1
2
3
4
5
6
7
8
9
10


Bernoulli's triangle falling diagonals

The table gives the th member, , of the th, , falling diagonal (0 for rightmost).

Bernoulli's triangle falling diagonals sequences
sequences A-number
0 {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, ...} A000079
1 {1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, ...} A000225
2 {1, 4, 11, 26, 57, 120, 247, 502, 1013, 2036, 4083, 8178, 16369, 32752, 65519, 131054, 262125, 524268, 1048555, 2097130, 4194281, 8388584, 16777191, 33554406, 67108837, ...} A000295
3 {1, 5, 16, 42, 99, 219, 466, 968, 1981, 4017, 8100, 16278, 32647, 65399, 130918, 261972, 524097, 1048365, 2096920, 4194050, 8388331, 16776915, 33554106, 67108512, 134217349, ...} A002662
4 {1, 6, 22, 64, 163, 382, 848, 1816, 3797, 7814, 15914, 32192, 64839, 130238, 261156, 523128, 1047225, 2095590, 4192510, 8386560, 16774891, 33551806, 67105912, 134214424, ...} A002663
5 {1, 7, 29, 93, 256, 638, 1486, 3302, 7099, 14913, 30827, 63019, 127858, 258096, 519252, 1042380, 2089605, 4185195, 8377705, 16764265, 33539156, 67090962, 134196874, 268411298, ...} A002664
6 {1, 8, 37, 130, 386, 1024, 2510, 5812, 12911, 27824, 58651, 121670, 249528, 507624, 1026876, 2069256, 4158861, 8344056, 16721761, 33486026, 67025182, 134116144, 268313018, ...} A035038
7 {1, 9, 46, 176, 562, 1586, 4096, 9908, 22819, 50643, 109294, 230964, 480492, 988116, 2014992, 4084248, 8243109, 16587165, 33308926, 66794952, 133820134, 267936278, 536249296, ...} A035039
8 {1, ...} A??????
9 {1, ...} A??????
10 {1, ...} A??????


Bernoulli's triangle falling diagonals formulae and values
Formulae

Generating

function

Differences

Partial sums

Partial sums of reciprocals

Sum of Reciprocals

0
1
2
3
4
5
6
7
8
9
10


Bernoulli's triangle central elements

Bernoulli's triangle central elements give the infinite sequence (Cf. A032443)

{1, 3, 11, 42, 163, 638, 2510, 9908, 39203, 155382, 616666, 2449868, 9740686, 38754732, 154276028, 614429672, 2448023843, 9756737702, 38897306018, 155111585372, 618679078298, ...}

which is given by the formula

The generating function is

Bernoulli's triangle (slope 1/2) rising diagonals

(...)

Bernoulli's triangle (slope 1/2) rising diagonals sums

(...)

Bernoulli's triangle (slope 1/2) rising diagonals alternating sign sums

(...)

Bernoulli's triangle (slope -1/2) falling diagonals

(...)

Bernoulli's triangle (slope -1/2) falling diagonals sums

(...)

Bernoulli's triangle (slope -1/2) falling diagonals alternating sign sums

(...)

See also

  • A?????? Multiplicative encoding of Bernoulli's triangle: Product p(i+1)^T(n,i).