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(1,2)-Pascal triangle

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The (1,2)-Pascal triangle (i.e. Lucas triangle) has its rightmost nonzero entries initialized to 2 and its leftmost nonzero entries (except the first row for n = 0) initialized to 1. Thus the rows of the (1,2)-Pascal triangle are the left-right reversal of the rows of the (2,1)-Pascal triangle, with the exception of the first row (for n = 0) which is now 2 instead of 1.

The (1,2)-Pascal triangle is a geometric arrangement of numbers produced recursively which generates (in its falling interior diagonals starting from the rightmost one) the square gnomonic numbers (the odd numbers,) the square numbers, the square pyramidal numbers and then the square hyperpyramidal numbers for dimension greather than 3 (The (2,1)-Pascal triangle , for the rectangular version, will generate those in its columns.) The original Pascal triangle, which is thus the (1,1)-Pascal triangle, generates (both in its columns for the rectangular version and in its falling interior diagonals starting from the rightmost one, since the (1,1)-Pascal triangle is symmetrical) the triangular gnomonic numbers (the natural numbers,) the triangular numbers], the tetrahedral numbers (the triangular pyramidal numbers) and then the hypertetrahedral numbers (the triangular hyperpyramidal numbers) for dimension greather than 3.

The rectangular version of the (1,2)-Pascal triangle
(figurate number triangle)
[1]
n = 0 2
1 1 2
2 1 3 2
3 1 4 5 2
4 1 5 9 7 2
5 1 6 14 16 9 2
6 1 7 20 30 25 11 2
7 1 8 27 50 55 36 13 2
8 1 9 35 77 105 91 49 15 2
9 1 10 44 112 182 196 140 64 17 2
10 1 11 54 156 294 378 336 204 81 19 2
11 1 12 65 210 450 672 714 540 285 100 21 2
12 1 13 77 275 660 1122 1386 1254 825 385 121 23 2
j = 0 1 2 3 4 5 6 7 8 9 10 11 12
(1+2x) (1+x)^(n-1)
or
"The rectangular version of the (1,2)-Pascal polynomials triangle"[2]
n = 0 2
1 1x + 2
2 1x 2 + 3x + 2
3 1x 3 + 4x 2 + 5x 2
4 1x 4 + 5x 3 + 9x 2 + 7x + 2
5 1x 5 + 6x 4 + 14x 3 + 16x 2 + 9x + 2
6 1x 6 + 7x 5 + 20x 4 + 30x 3 + 25x 2 + 11x + 2
7 1x 7 + 8x 6 + 27x 5 + 50x 4 + 55x 3 + 36x 2 + 13x + 2
8 1x 8 + 9x 7 + 35x 6 + 77x 5 + 105x 4 + 91x 3 + 49x 2 + 15x + 2
9 1x 9 + 10x 8 + 44x 7 + 112x 6 + 182x 5 + 196x 4 + 140x 3 + 64x 2 + 17x + 2
10 1x 10 + 11x 9 + 54x 8 + 156x 7 + 294x 6 + 378x 5 + 336x 4 + 204x 3 + 81x 2 + 19x + 2
j = 0 1 2 3 4 5 6 7 8 9 10

In the equilateral version of the (1,2)-Pascal triangle, we start with a cell (row 0) initialized to 2, with all the leftmost nonzero cells in the rows below initialized to 1, in a staggered array of empty (0) cells. We then recursively evaluate the cells as the sum of the two cells staggered above. The triangle thus grows into an equilateral triangle.

In the rectangular version of the (1,2)-Pascal triangle, we start with a cell (row 0) initialized to 2, with all the cells below it initialized to 1, in a regular array of empty (0) cells. We then recursively evaluate the cells as the sum of the one above left and the one directly above. The triangle thus grows into a rectangular triangle.

The rightmost nonzero cells on each rows are therefore set to 2+0 = 2. The leftmost nonzero cells on each rows except the first one (for n = 0) are initialized to 1. All the interior cells are necessarily greater than 2. The number of cells from rows 0 to n which are equal to 1 is n. (Cf. A001477(n),) the number of cells from rows 0 to n which are equal to 2 is n+1 (Cf. A001477(n+1),) and the number of cells from rows 0 to n which are greater than or equal to 3 is , the (n-1)th triangular number.

Recursion rule

The (1,2)-Pascal triangle recursion rule is:

Formulae

where when n < 0, r < 0 or n - r < 0,[3] and is cell (n, j) of Pascal's triangle.

(1,2)-Pascal triangle rows

n = 0 2
1 1 2
2 1 3 2
3 1 4 5 2
4 1 5 9 7 2
5 1 6 14 16 9 2
6 1 7 20 30 25 11 2
7 1 8 27 50 55 36 13 2
8 1 9 35 77 105 91 49 15 2
9 1 10 44 112 182 196 140 64 17 2
10 1 11 54 156 294 378 336 204 81 19 2
11 1 12 65 210 450 672 714 540 285 100 21 2
12 1 13 77 275 660 1122 1386 1254 825 385 121 23 2
j = 0 1 2 3 4 5 6 7 8 9 10 11 12


The (1,2)-Pascal triangle rows give an infinite sequence of finite sequences:

{{2}, {1, 2}, {1, 3, 2}, {1, 4, 5, 2}, {1, 5, 9, 7, 2}, {1, 6, 14, 16, 9, 2}, {1, 7, 20, 30, 25, 11, 2}, {1, 8, 27, 50, 55, 36, 13, 2}, {1, 9, 35, 77, 105, 91, 49, 15, 2}, ...}

The generating function for the jth, j ≥ 0, member of the nth, n ≥ 1, subsequence is:

The concatenation of the infinite sequence of finite sequences gives the infinite sequence (Cf. A029635(n) with a(0) corrected to 2):

{2, 1, 2, 1, 3, 2, 1, 4, 5, 2, 1, 5, 9, 7, 2, 1, 6, 14, 16, 9, 2, 1, 7, 20, 30, 25, 11, 2, 1, 8, 27, 50, 55, 36, 13, 2, 1, 9, 35, 77, 105, 91, 49, 15, 2, ...}

The generating function for the ith, i ≥ 0, member is:

(1,2)-Pascal triangle rows sums

The sums of the respective finite sequences give the infinite sequence (Cf. A042950(n)):

{2, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, ...}

with members given by the formula:

where:

The generating function is:

(1,2)-Pascal triangle rows alternating sign sums

The alternating sign sums of the respective finite sequences give the infinite sequence (Cf. A??????):

{2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...}

with members given by the formula:

where:

The generating function is:

(1,2)-Pascal (rectangular) triangle columns (and Chebyshev polynomials?)

n = 0 2
1 1 2
2 1 3 2
3 1 4 5 2
4 1 5 9 7 2
5 1 6 14 16 9 2
6 1 7 20 30 25 11 2
7 1 8 27 50 55 36 13 2
8 1 9 35 77 105 91 49 15 2
9 1 10 44 112 182 196 140 64 17 2
10 1 11 54 156 294 378 336 204 81 19 2
11 1 12 65 210 450 672 714 540 285 100 21 2
12 1 13 77 275 660 1122 1386 1254 825 385 121 23 2
j = 0 1 2 3 4 5 6 7 8 9 10 11 12


The first column (for j = 0) gives the sequence (Cf. A054977(n)):

{2, 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, ...}

with members given by the formula:

where:

The generating function is:

This happens to be the continued fraction expansion (2 being the integral part) of where is the Golden ratio.

The second column (for j = 1) gives the nonunit positive integers (Cf. A000027(n+1), n ≥ 1).

Table of columns sequences

The i th, i ≥ 0, member of column j appears in row j+i.

(1,2)-Pascal triangle columns sequences
j sequences A-number
0 {2, 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, ...} A054977(i)
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, ...} A000027(i+2)
2 {2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, ...} A000096(i+1)
3 {2, 7, 16, 30, 50, 77, 112, 156, 210, 275, 352, 442, 546, 665, 800, 952, 1122, 1311, 1520, 1750, 2002, 2277, 2576, 2900, 3250, 3627, 4032, 4466, 4930, 5425, ...} A005581(i+2)
4 {2, 9, 25, 55, 105, 182, 294, 450, 660, 935, 1287, 1729, 2275, 2940, 3740, 4692, 5814, 7125, 8645, 10395, 12397, 14674, 17250, 20150, 23400, 27027, 31059, ...} A005582(i+1)
5 {2, 11, 36, 91, 196, 378, 672, 1122, 1782, 2717, 4004, 5733, 8008, 10948, 14688, 19380, 25194, 32319, 40964, 51359, 63756, 78430, 95680, 115830, 139230, ...} A005583(i+1)
6 {2, 13, 49, 140, 336, 714, 1386, 2508, 4290, 7007, 11011, 16744, 24752, 35700, 50388, 69768, 94962, 127281, 168245, 219604, 283360, 361790, 457470, ...} A005584(i+1)
7 {2, 15, 64, 204, 540, 1254, 2640, 5148, 9438, 16445, 27456, 44200, 68952, 104652, 155040, 224808, 319770, 447051, 615296, 834900, 1118260, 1480050, ...} A??????
8 {2, 17, 81, 285, 825, 2079, 4719, 9867, 19305, 35750, 63206, 107406, 176358, 281010, 436050, 660858, 980628, 1427679, 2042975, 2877875, 3996135, ...} A??????
9 {2, 19, 100, 385, 1210, 3289, 8008, 17875, 37180, 72930, 136136, 243542, 419900, 700910, 1136960, 1797818, 2778446, 4206125, 6249100, 9126975, ...} A??????
10 {2, 21, 121, 506, 1716, 5005, 13013, 30888, 68068, 140998, 277134, 520676, 940576, 1641486, 2778446, 4576264, 7354710, 11560835, 17809935, 26936910, ...} A??????
11 {2, 23, 144, 650, 2366, 7371, 20384, 51272, 119340, 260338, 537472, 1058148, 1998724, 3640210, 6418656, 10994920, 18349630, 29910465, 47720400, ...} A??????
12 {2, 25, 169, 819, 3185, 10556, 30940, 82212, 201552, 461890, 999362, 2057510, 4056234, 7696444, 14115100, 25110020, 43459650, 73370115, 121090515, ...} A??????


Table of columns sequences related formulae

The i th, i ≥ 0, member of column j appears in row j+i.

(1,2)-Pascal triangle columns sequences related formulae
j Formulae


Generating

function

for i th (i ≥ 0)

member of column


Order

of basis

Differences

Partial sums

Partial sums of reciprocals

Sum of Reciprocals[4][5]

0
1



(for )



2


3


4
5
6
7
8
9
10
11
12


(1,2)-Pascal (rectangular) triangle falling diagonals and square (hyper)pyramidal numbers

(Square Hyperpyramidal) Figurate Number Triangle
n = 0 2
1 1 2
2 1 3 2
3 1 4 5 2
4 1 5 9 7 2
5 1 6 14 16 9 2
6 1 7 20 30 25 11 2
7 1 8 27 50 55 36 13 2
8 1 9 35 77 105 91 49 15 2
9 1 10 44 112 182 196 140 64 17 2
10 1 11 54 156 294 378 336 204 81 19 2
11 1 12 65 210 450 672 714 540 285 100 21 2
12 1 13 77 275 660 1122 1386 1254 825 385 121 23 2
j = 0 1 2 3 4 5 6 7 8 9 10 11 12


The partial sums of the dth (starting from the right with d = 0 giving the 2's diagonal) falling diagonal, d ≥ 1, build the entries of the (d+1)th (starting from the right) falling diagonal, thus begetting the (d+1)-dimensional square (hyper)pyramidal numbers from the d-dimensional ones:

The dth falling diagonal, d ≥ 1, gives the d-dimensional square hyperpyramidal numbers, forming square (hyper)pyramids, e.g.:

d=1 1-dimensional square hyperpyramidal numbers Square gnomon numbers (2 0D-cells facets) (Square gnomons)
d=2 2-dimensional square hyperpyramidal numbers Square numbers (4 1D-cells facets) (Squares)
d=3 3-dimensional square hyperpyramidal numbers Square pyramidal numbers (5 2D-cells facets) (Square pyramids)
d=4 4-dimensional square hyperpyramidal numbers Square 4D-hyperpyramidal numbers (? 3D-cells facets) (Square 4D-hyperpyramids)
d=5 5-dimensional square hyperpyramidal numbers Square 5D-hyperpyramidal numbers (? 4D-cells facets) (Square 5D-hyperpyramids)
d=6 6-dimensional square hyperpyramidal numbers Square 6D-hyperpyramidal numbers (? 5D-cells facets) (Square 6D-hyperpyramids)
d=7 7-dimensional square hyperpyramidal numbers Square 7D-hyperpyramidal numbers (? 6D-cells facets) (Square 7D-hyperpyramids)
d=8 8-dimensional square hyperpyramidal numbers Square 8D-hyperpyramidal numbers (? 7D-cells facets) (Square 8D-hyperpyramids)

where (-1D)-cells correspond to the empty set, 0D-cells are vertices, 1D-cells are edges, 2D-cells are faces, and so on...

Table of falling diagonals sequences

The falling diagonals sequences of the (1,2)-Pascal triangle correspond to the columns sequences of the (2,1)-Pascal triangle, except that the apex being 2, the 0th member of the 0th falling diagonal is 2 (instead of 1 for the 0th member of the 0th column of the (2,1)-Pascal triangle.)

The i th, i ≥ 0, member of falling diagonal d appears in row d+i.

(1,2)-Pascal triangle falling diagonals sequences
d Sequence A-number
0 {2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...} A007395(i+1)


Table of falling diagonals sequences related formulae

The falling diagonals sequences related formulae of the (1,2)-Pascal triangle correspond to the columns sequences related formulae of the (2,1)-Pascal triangle, except that the apex being 2, the 0th member of the 0th falling diagonal is 2 (instead of 1 for the 0th member of the 0th column of the (2,1)-Pascal triangle.)

The i th, i ≥ 0, member of falling diagonal d appears in row d+i.

(1,2)-Pascal triangle falling diagonals sequences related formulae
d Formulae




Generating

function

for i th (i ≥ 0)

column member



Order

of basis

Differences

Partial sums

Partial sums of reciprocals

Sum of Reciprocals[6][7]

0


(1,2)-Pascal (rectangular) triangle rising diagonals

Lucas triangle
n = 0 2
1 1 2
2 1 3 2
3 1 4 5 2
4 1 5 9 7 2
5 1 6 14 16 9 2
6 1 7 20 30 25 11 2
7 1 8 27 50 55 36 13 2
8 1 9 35 77 105 91 49 15 2
9 1 10 44 112 182 196 140 64 17 2
10 1 11 54 156 294 378 336 204 81 19 2
11 1 12 65 210 450 672 714 540 285 100 21 2
12 1 13 77 275 660 1122 1386 1254 825 385 121 23 2
j = 0 1 2 3 4 5 6 7 8 9 10 11 12


The rising diagonals (starting with the 0th diagonal) give an infinite sequence of finite sequences (which are the coefficients of Lucas polynomials (or Cardan polynomials):)[8]

{{2}, {1}, {1, 2}, {1, 3}, {1, 4, 2}, {1, 5, 5}, {1, 6, 9, 2}, {1, 7, 14, 7}, {1, 8, 20, 16, 2}, {1, 9, 27, 30, 9}, {1, 10, 35, 50, 25, 2}, ...}

The concatenated coefficients of Lucas polynomials (or Cardan polynomials) give the infinite sequence (cf. A034807(n)):

{2, 1, 1, 2, 1, 3, 1, 4, 2, 1, 5, 5, 1, 6, 9, 2, 1, 7, 14, 7, 1, 8, 20, 16, 2, 1, 9, 27, 30, 9, 1, 10, 35, 50, 25, 2, ...}

(1,2)-Pascal (rectangular) triangle rising diagonals sums and Lucas numbers

The sums of the respective rising diagonals give:

{2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, ...}

which are the mth Lucas numbers (Cf. A000032(m),) and may be obtained using the Binet formula:

where and are the roots of and where is the Golden ratio.

The generating function is:

Compare with the rising diagonals sums of the (2,1)-Pascal triangle, which give the Fibonacci numbers.

(1,2)-Pascal triangle central elements

The central elements (for row 2m, m ≥ 0) of the (1,2)-Pascal triangle give the sequence (Cf. A029651(m), except for m = 0 where we now have 2 (instead of 1 for the (2,1)-Pascal triangle central elements):

{2, 3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148, 4056234, 15600900, 60174900, 232676280, 901620585, 3500409330, 13612702950, 53017895700, ...}

which is given by the formulae:

or:

where:

is the mth Catalan number (also called Segner number) (Cf. A000108(m).)

The generating function is:

where is the generating function of the Catalan numbers:

See also

  • A029635 The (1,2)-Pascal triangle (or Lucas triangle) read by rows (where a(0) has been corrected to 2.)
  • A123558 Multiplicative encoding of the (1,2)-Pascal triangle (or Lucas triangle) A029635.
  • A114525 Triangle of coefficients of the Lucas (w-)polynomials.



Notes

  1. Weisstein, Eric W., Figurate Number Triangle, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/FigurateNumberTriangle.html]
  2. Compare with the Lucas polynomials.
  3. Weisstein, Eric W., Binomial Coefficient, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/BinomialCoefficient.html]
  4. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  5. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
  6. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  7. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.
  8. Weisstein, Eric W., Lucas Polynomial, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/LucasPolynomial.html]

External links