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 $\scriptstyle n\,=\,3\,$ 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 $\scriptstyle n\,=\,3\,$ 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).

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

$T(n,0)=1,\,$

$T(n,n)=2^{n},\,$

$T(n,k)=T(n-1,k-1)+T(n-1,k),\ 0<k<n.\,$

Bernoulli's triangle formula

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

and

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

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$\scriptstyle (n),\,n\,\geq \,0\,$)

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

{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

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

The generating function is

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

Bernoulli's triangle rows alternating sign sums

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

Bernoulli's triangle rising diagonals

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

Bernoulli's triangle rising diagonals sequences

$i\,$	$T(i+j,i),\ j\geq 0\,$ 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($\scriptstyle j\,$)
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($\scriptstyle j+2\,$)
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($\scriptstyle j+2\,$)
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($\scriptstyle j+3\,$)
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($\scriptstyle j+5\,$)
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($\scriptstyle j+5\,$)
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($\scriptstyle j+6\,$)
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($\scriptstyle j+7\,$)
8	{256, 511, 1013, 1981, 3797, 7099, 12911, 22819, 39203, 65536, 106762, 169766, 263950, 401930, 600370, 880970, 1271626, 1807781, 2533987, 3505699, 4791323, 6474541, ...}	A008861($\scriptstyle j+8\,$)
9	{512, 1023, 2036, 4017, 7814, 14913, 27824, 50643, 89846, 155382, 262144, 431910, 695860, 1097790, 1698160, 2579130, 3850756, 5658537, 8192524, 11698223, 16489546, ...}	A008862($\scriptstyle j+9\,$)
10	{1024, 2047, 4083, 8100, 15914, 30827, 58651, 109294, 199140, 354522, 616666, 1048576, 1744436, 2842226, 4540386, 7119516, 10970272, 16628809, 24821333, 36519556, ...}	A008863($\scriptstyle j+10\,$)

Bernoulli's triangle rising diagonals formulae and values

$i\,$	Formulae $T(i+j,i)=\,$ $\sum _{k=0}^{i}{\binom {n}{k}}\,$	Generating function $G_{\{T(i+j,i)\}}(x)=\,$ ${\frac {1}{x^{i}}}{\bigg \{}\sum _{k=0}^{i}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{i}-1}{2x-1}}{\bigg \}}\,$	Differences $T(i+j,i)-T(i+j-1,i)=\,$	Partial sums $\sum _{j=0}^{m}{T(i+j,i)}=\,$	Partial sums of reciprocals $\sum _{j=0}^{m}{1 \over {T(i+j,i)}}=\,$	Sum of Reciprocals $\sum _{j=0}^{\infty }{1 \over {T(i+j,i)}}=\,$
0	$\sum _{k=0}^{0}{\binom {n}{k}}\,$	${\frac {1}{x^{0}}}{\bigg \{}\sum _{k=0}^{0}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{0}-1}{2x-1}}{\bigg \}}\,$
1	$\sum _{k=0}^{1}{\binom {n}{k}}\,$	${\frac {1}{x^{1}}}{\bigg \{}\sum _{k=0}^{1}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{1}-1}{2x-1}}{\bigg \}}\,$
2	$\sum _{k=0}^{2}{\binom {n}{k}}\,$	${\frac {1}{x^{2}}}{\bigg \{}\sum _{k=0}^{2}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{2}-1}{2x-1}}{\bigg \}}\,$
3	$\sum _{k=0}^{3}{\binom {n}{k}}\,$	${\frac {1}{x^{3}}}{\bigg \{}\sum _{k=0}^{3}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{3}-1}{2x-1}}{\bigg \}}\,$
4	$\sum _{k=0}^{4}{\binom {n}{k}}\,$	${\frac {1}{x^{4}}}{\bigg \{}\sum _{k=0}^{4}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{4}-1}{2x-1}}{\bigg \}}\,$
5	$\sum _{k=0}^{5}{\binom {n}{k}}\,$	${\frac {1}{x^{5}}}{\bigg \{}\sum _{k=0}^{5}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{5}-1}{2x-1}}{\bigg \}}\,$
6	$\sum _{k=0}^{6}{\binom {n}{k}}\,$	${\frac {1}{x^{6}}}{\bigg \{}\sum _{k=0}^{6}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{6}-1}{2x-1}}{\bigg \}}\,$
7	$\sum _{k=0}^{7}{\binom {n}{k}}\,$	${\frac {1}{x^{7}}}{\bigg \{}\sum _{k=0}^{7}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{7}-1}{2x-1}}{\bigg \}}\,$
8	$\sum _{k=0}^{8}{\binom {n}{k}}\,$	${\frac {1}{x^{8}}}{\bigg \{}\sum _{k=0}^{8}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{8}-1}{2x-1}}{\bigg \}}\,$
9	$\sum _{k=0}^{9}{\binom {n}{k}}\,$	${\frac {1}{x^{9}}}{\bigg \{}\sum _{k=0}^{9}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{9}-1}{2x-1}}{\bigg \}}\,$
10	$\sum _{k=0}^{10}{\binom {n}{k}}\,$	${\frac {1}{x^{10}}}{\bigg \{}\sum _{k=0}^{10}{\frac {x^{k}}{(1-x)^{k+1}}}-{\frac {(2x)^{10}-1}{2x-1}}{\bigg \}}\,$

Bernoulli's triangle falling diagonals

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

Bernoulli's triangle falling diagonals sequences

$i\,$	$T(i+j,j),\ j\geq 0\,$ 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$\scriptstyle (j)\,$
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$\scriptstyle (j+1)\,$
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$\scriptstyle (j+2)\,$
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$\scriptstyle (j+3)\,$
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$\scriptstyle (j+4)\,$
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$\scriptstyle (j+5)\,$
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$\scriptstyle (j+6)\,$
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$\scriptstyle (j+7)\,$
8	{1, ...}	A??????
9	{1, ...}	A??????
10	{1, ...}	A??????

Bernoulli's triangle falling diagonals formulae and values

$i\,$	Formulae $T(i+j,j)=\,$	Generating function $G_{\{T(i+j,j)\}}(x)=\,$	Differences $T(i+j,j)-T(i+j-1,j)=\,$	Partial sums $\sum _{j=0}^{m}{T(i+j,j)}=\,$	Partial sums of reciprocals $\sum _{j=0}^{m}{1 \over {T(i+j,j)}}=\,$	Sum of Reciprocals $\sum _{j=0}^{\infty }{1 \over {T(i+j,j)}}=\,$
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$\scriptstyle (n),n\,\geq \,0\,$)

{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

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

The generating function is

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

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

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

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

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

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

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

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See also

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