This site is supported by donations to The OEIS Foundation.

# Bernoulli's triangle

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 ${\displaystyle \scriptstyle n\,=\,3\,}$ 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 ${\displaystyle \scriptstyle n\,=\,3\,}$ 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).

 ${\displaystyle n\,}$ B E R N O U L L I ' S T R I A N G L E Row sums ${\displaystyle \sum _{k=0}^{n}T(n,k),\,}$ ${\displaystyle (n+2)\,2^{n-1}\,}$ (A001792(${\displaystyle \scriptstyle n\,}$)) 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 ${\displaystyle \scriptstyle a(5)\,}$: Maximal number of regions obtained by joining ${\displaystyle \scriptstyle n\,}$ points around a circle by straight lines. 6144 11 1 12 67 A000125 starting with ${\displaystyle \scriptstyle a(3)\,}$: Cake numbers: maximal number of pieces resulting from ${\displaystyle \scriptstyle n\,}$ planar cuts through a cube or cake. 13312 12 1 13 A000124 starting with ${\displaystyle \scriptstyle a(2)\,}$: Central polygonal numbers 28672 13 1 A000027 starting with ${\displaystyle \scriptstyle a(2)\,}$: 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 ${\displaystyle \scriptstyle 2^{n}\,}$, i.e. the row sums of Pascal's triangle. Then a recurrence equation identical to Pascal's triangle is applied.

${\displaystyle T(n,0)=1,\,}$
${\displaystyle T(n,n)=2^{n},\,}$
${\displaystyle T(n,k)=T(n-1,k-1)+T(n-1,k),\ 0

## Bernoulli's triangle formula

${\displaystyle T(n,k)=\sum _{i=0}^{k}{\binom {n}{i}}\,}$

and

${\displaystyle T(n,k)-T(n,k-1)={\binom {n}{k}}\,}$

## 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${\displaystyle \scriptstyle (n),\,n\,\geq \,0\,}$)

{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${\displaystyle \scriptstyle (n),\,n\,\geq \,0\,}$)

{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

${\displaystyle \sum _{k=0}^{n}T(n,k)=(n+2)\ 2^{n-1}=n\ 2^{n-1}+2^{n}.\,}$

The generating function is

${\displaystyle G_{\{\sum _{k=0}^{n}T(n,k)\}}(x)={\frac {1-x}{(1-2x)^{2}}}.\,}$

### Bernoulli's triangle rows alternating sign sums

${\displaystyle \sum _{k=0}^{n}(-1)^{k}\ T(n,k)=?\,}$

## Bernoulli's triangle rising diagonals

The table gives the ${\displaystyle \scriptstyle j\,}$th member, ${\displaystyle \scriptstyle j\,\geq \,0\,}$, of the ${\displaystyle \scriptstyle i\,}$th, ${\displaystyle \scriptstyle i\,\geq \,0\,}$, rising diagonal (0 for leftmost).

Bernoulli's triangle rising diagonals sequences
${\displaystyle i\,}$ ${\displaystyle T(i+j,i),\ j\geq 0\,}$ 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(${\displaystyle \scriptstyle j\,}$)
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(${\displaystyle \scriptstyle j+2\,}$)
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(${\displaystyle \scriptstyle j+2\,}$)
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(${\displaystyle \scriptstyle j+3\,}$)
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(${\displaystyle \scriptstyle j+5\,}$)
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(${\displaystyle \scriptstyle j+5\,}$)
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(${\displaystyle \scriptstyle j+6\,}$)
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(${\displaystyle \scriptstyle j+7\,}$)
8 {256, 511, 1013, 1981, 3797, 7099, 12911, 22819, 39203, 65536, 106762, 169766, 263950, 401930, 600370, 880970, 1271626, 1807781, 2533987, 3505699, 4791323, 6474541, ...} A008861(${\displaystyle \scriptstyle j+8\,}$)
9 {512, 1023, 2036, 4017, 7814, 14913, 27824, 50643, 89846, 155382, 262144, 431910, 695860, 1097790, 1698160, 2579130, 3850756, 5658537, 8192524, 11698223, 16489546, ...} A008862(${\displaystyle \scriptstyle j+9\,}$)
10 {1024, 2047, 4083, 8100, 15914, 30827, 58651, 109294, 199140, 354522, 616666, 1048576, 1744436, 2842226, 4540386, 7119516, 10970272, 16628809, 24821333, 36519556, ...} A008863(${\displaystyle \scriptstyle j+10\,}$)

Bernoulli's triangle rising diagonals formulae and values
${\displaystyle i\,}$ Formulae

${\displaystyle T(i+j,i)=\,}$

${\displaystyle \sum _{k=0}^{i}{\binom {n}{k}}\,}$

Generating

function

${\displaystyle G_{\{T(i+j,i)\}}(x)=\,}$

${\displaystyle {\frac {1}{x^{i}}}{\bigg \{}\sum _{k=0}^{i}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{i}-1}{2x-1}}{\bigg \}}\,}$

Differences

${\displaystyle T(i+j,i)-T(i+j-1,i)=\,}$

Partial sums

${\displaystyle \sum _{j=0}^{m}{T(i+j,i)}=\,}$

Partial sums of reciprocals

${\displaystyle \sum _{j=0}^{m}{1 \over {T(i+j,i)}}=\,}$

Sum of Reciprocals

${\displaystyle \sum _{j=0}^{\infty }{1 \over {T(i+j,i)}}=\,}$

0 ${\displaystyle \sum _{k=0}^{0}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{0}}}{\bigg \{}\sum _{k=0}^{0}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{0}-1}{2x-1}}{\bigg \}}\,}$
1 ${\displaystyle \sum _{k=0}^{1}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{1}}}{\bigg \{}\sum _{k=0}^{1}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{1}-1}{2x-1}}{\bigg \}}\,}$
2 ${\displaystyle \sum _{k=0}^{2}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{2}}}{\bigg \{}\sum _{k=0}^{2}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{2}-1}{2x-1}}{\bigg \}}\,}$
3 ${\displaystyle \sum _{k=0}^{3}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{3}}}{\bigg \{}\sum _{k=0}^{3}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{3}-1}{2x-1}}{\bigg \}}\,}$
4 ${\displaystyle \sum _{k=0}^{4}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{4}}}{\bigg \{}\sum _{k=0}^{4}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{4}-1}{2x-1}}{\bigg \}}\,}$
5 ${\displaystyle \sum _{k=0}^{5}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{5}}}{\bigg \{}\sum _{k=0}^{5}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{5}-1}{2x-1}}{\bigg \}}\,}$
6 ${\displaystyle \sum _{k=0}^{6}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{6}}}{\bigg \{}\sum _{k=0}^{6}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{6}-1}{2x-1}}{\bigg \}}\,}$
7 ${\displaystyle \sum _{k=0}^{7}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{7}}}{\bigg \{}\sum _{k=0}^{7}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{7}-1}{2x-1}}{\bigg \}}\,}$
8 ${\displaystyle \sum _{k=0}^{8}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{8}}}{\bigg \{}\sum _{k=0}^{8}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{8}-1}{2x-1}}{\bigg \}}\,}$
9 ${\displaystyle \sum _{k=0}^{9}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{9}}}{\bigg \{}\sum _{k=0}^{9}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{9}-1}{2x-1}}{\bigg \}}\,}$
10 ${\displaystyle \sum _{k=0}^{10}{\binom {n}{k}}\,}$ ${\displaystyle {\frac {1}{x^{10}}}{\bigg \{}\sum _{k=0}^{10}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{10}-1}{2x-1}}{\bigg \}}\,}$

## Bernoulli's triangle falling diagonals

The table gives the ${\displaystyle \scriptstyle j\,}$th member, ${\displaystyle \scriptstyle j\,\geq \,0\,}$, of the ${\displaystyle \scriptstyle i\,}$th, ${\displaystyle \scriptstyle i\,\geq \,0\,}$, falling diagonal (0 for rightmost).

Bernoulli's triangle falling diagonals sequences
${\displaystyle i\,}$ ${\displaystyle T(i+j,j),\ j\geq 0\,}$ 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${\displaystyle \scriptstyle (j)\,}$
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${\displaystyle \scriptstyle (j+1)\,}$
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${\displaystyle \scriptstyle (j+2)\,}$
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${\displaystyle \scriptstyle (j+3)\,}$
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${\displaystyle \scriptstyle (j+4)\,}$
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${\displaystyle \scriptstyle (j+5)\,}$
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${\displaystyle \scriptstyle (j+6)\,}$
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${\displaystyle \scriptstyle (j+7)\,}$
8 {1, ...} A??????
9 {1, ...} A??????
10 {1, ...} A??????

Bernoulli's triangle falling diagonals formulae and values
${\displaystyle i\,}$ Formulae

${\displaystyle T(i+j,j)=\,}$

Generating

function

${\displaystyle G_{\{T(i+j,j)\}}(x)=\,}$

Differences

${\displaystyle T(i+j,j)-T(i+j-1,j)=\,}$

Partial sums

${\displaystyle \sum _{j=0}^{m}{T(i+j,j)}=\,}$

Partial sums of reciprocals

${\displaystyle \sum _{j=0}^{m}{1 \over {T(i+j,j)}}=\,}$

Sum of Reciprocals

${\displaystyle \sum _{j=0}^{\infty }{1 \over {T(i+j,j)}}=\,}$

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${\displaystyle \scriptstyle (n),n\,\geq \,0\,}$)

{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

${\displaystyle T(2n,n)=\sum _{i=0}^{n}{\binom {2n}{i}}.\,}$

The generating function is

${\displaystyle G_{\{T(2n,n)\}}(x)\equiv \sum _{n=0}^{\infty }T(2n,n)\,x^{n}=\ ?\,}$

(...)

(...)

(...)

(...)

(...)

(...)