A109649 -id:A109649 - cp's OEIS frontend

A109672 Entries in 3-dimensional solids related to Prouhet-Tarry problem.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 3, 3, 1, 1, 2, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 1, 1, 2, 5, 2, 1, 5, 5, 1, 1, 2, 1, 1, 3, 3, 3, 6, 3, 1, 3, 3, 1, 1, 4, 6, 4, 1, 4, 12, 12, 4, 6, 12, 6, 4, 4, 1, 1, 3, 3, 1, 1, 7, 12, 7, 1, 3, 12

Offset: 0

Philippe Deléham, Aug 07 2005

Table of slices [n,k] of solids, read by antidiagonals, each slice [n,k] read by rows.
Slice [n,0] gives A046816.
Slice [0,k] gives A109649.
Slice [n,n] gives A109673.

			Slice [0,0]:
...1...
Slice [0,1]:
... 1 1 ...
.... 1 ....
Slice [1,0]:
.... 1 ....
... 1 1...
Slice [0,2]:
.. 1 2 1 ...
.... 2 2 ...
..... 1 .....
Slice [1,1]:
... 1 1 ...
.. 1 3 1..
... 1 1 ...
Slice [2,0]:
..... 1 .....
.... 2 2 ...
.. 1 2 1 ...
Slice [0,3]:
.. 1 3 3 1 ...
... 3 6 3 ....
.... 3 3 ......
..... 1 ........
Slice [1,2]:
... 1 2 1 ...
.. 1 5 5 1 ...
... 2 5 2 ...
.... 1 1 ...
Slice [2,1]:
.... 1 1 ...
... 2 5 2 ...
.. 1 5 5 1 ...
... 1 2 1 ...
Slice [3,0]:
..... 1 .....
.... 3 3 ....
... 3 6 3 ...
.. 1 3 3 1 ...

Sum of terms in 2D slice [n, k] is 3^(n+k); example : 1+2+1+1+5+5+1+2+5+2+1+127=3^(2+1) for slice [1, 2].

A109390 Entries in 3-dimensional solid related to Prouhet-Tarry problem.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 5, 2, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 7, 3, 3, 12, 12, 3, 1, 7, 12, 7, 1, 1, 3, 3, 1, 1, 1, 4, 9, 4, 6, 22, 22, 6, 4, 22, 36, 22, 4, 1, 9, 22, 22, 9, 1, 1, 1, 5, 11, 5, 10, 35, 35, 10, 10, 50, 80, 50, 10, 5, 35, 80, 80, 35, 5, 1, 11, 35, 50

Offset: 0

Philippe Deléham, Aug 26 2005

Entries of slices [n,1] in A109672, read by rows.

Slice [n,0] gives A046816, slice [0,k] gives A109649, slice [n,n] gives A109673, slice [1,k] gives A109393.

			Slice [0,1]:
... 1 1 ...
.... 1 ....
Slice [1,1]:
... 1 1 ...
.. 1 3 1 ...
... 1 1 ...
Slice [2,1]:
.... 1 1 ....
... 2 5 2 ...
.. 1 5 5 1 ...
... 1 2 1 ...
Slice [3,1]:
..... 1 1 .....
.... 3 7 3 ....
... 3 12 12 3 ...
.. 1 7 12 7 1 ...
... 1 3 3 1 ...
Slice [4,1]:
...... 1 1 ......
..... 4 9 4 .....
.... 6 22 22 6 ....
... 4 22 36 22 4 ...
.. 1 9 22 22 9 1 ...
... 1 4 6 4 1 ...
Slice [5,1]:
....... 1 1 .......
...... 5 11 5 ......
..... 10 35 35 10 .....
.... 10 50 80 50 10 ....
... 5 35 80 80 35 5 ...
.. 1 11 35 50 35 11 1 ...
... 1 5 10 10 5 1 ...

Cf. A109393.

Sum of terms in 2D slice [n, 1] is 3^(n+1).

A109393 Entries in 3-dimensional solid related to Prouhet-Tarry problem.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 1, 3, 3, 1, 1, 7, 12, 7, 1, 3, 12, 12, 3, 3, 7, 3, 1, 1, 1, 4, 6, 4, 1, 1, 9, 22, 22, 9, 1, 4, 22, 36, 22, 4, 6, 22, 22, 6, 4, 9, 4, 1, 1, 1, 5, 10, 10, 5, 1, 1, 11, 35, 50, 35, 11, 1, 5, 35, 80, 80, 35, 5, 10, 50, 80, 50

Offset: 0

Philippe Deléham, Aug 26 2005

Entries of slices [1,k] in A109672, read by rows.

Slice [n,0] gives A046816, slice [0,k] gives A109649, slice [n,n] gives A109673, slice [n,1] gives A109390.

			Slice [1,0]:
... 1 ...
.. 1 1 ...
Slice [1,1]:
... 1 1 ...
.. 1 3 1 ...
... 1 1 ...
Slice [1,2]:
... 1 2 1 ...
.. 1 5 5 1 ...
... 2 5 2 ...
.... 1 1 ....
Slice [1,3]:
... 1 3 3 1 ...
.. 1 7 12 7 1 ...
... 3 12 12 3 ...
.... 3 7 3 ....
..... 1 1 .....
Slice [1,4]:
... 1 4 6 4 1 ...
.. 1 9 22 22 9 1 ...
... 4 22 36 22 4 ...
.... 6 22 22 6 ....
..... 4 9 4 .....
...... 1 1 ......
Slice [1,5]:
... 1 5 10 10 5 1 ...
.. 1 11 35 50 35 11 1 ...
... 5 35 80 80 35 5 ...
.... 10 50 80 50 10 ....
..... 10 35 35 10 .....
...... 5 11 5 ......
....... 1 1

Cf. A109390.

Sum of terms in 2D slice [1, k] is 3^(1+k).

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 3, 3, 1, 3, 6, 3, 3, 3, 1, 1, 4, 4, 6, 12, 6, 4, 12, 12, 4, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 5, 20, 30, 20, 5, 10, 30, 30, 10, 10, 20, 10, 5, 5, 1, 1, 6, 6, 15, 30, 15, 20, 60, 60, 20, 15, 60, 90, 60, 15, 6, 30, 60, 60, 30, 6, 1, 6, 15, 20, 15, 6, 1

Offset: 0

Harlan J. Brothers, Jun 16 2010

P_h = level h of Pascal's prism where P_1 = Pascal's triangle (A007318) and P_2 = denominators of Leibniz harmonic triangle (A003506). A sequence of length k through P is defined by P for n = {1, 2, 3, ..., k}.

			Prism begins (levels 1-4):
1
1 1
1 2 1
1 3 3 1
1
2 2
3 6 3
4 12 12 4
1
3 3
6 12 6
10 30 30 10
1
4 4
10 20 10
20 60 60 20

H. J. Brothers, Pascal's prism, The Mathematical Gazette, 96 (July 2012), 213-220.
H. J. Brothers, Pascal's Prism: Supplementary Material

Level 1 = A007318.
Level 2 = A003506.
Level 3 = A094305.
Level 4 = A178820.
Level 5 = A178821.
Level 6 = A178822.
Sums of shallow diagonals for each level correspond to rows of square A037027.
Contains A109649 and A046816.
P = A000984.
P = A006480.
P = A000897.
P<3n-2, 3n-2, n> = A113424.

end = 5; Column/@Table[Multinomial[h, i-j, j], {h, 0, end}, {i, 0, end}, {j, 0, i}]

a_(h, i, j) = (h+i-2; h-1, i-j, j-1), h >= 1, i >= 1, 1 <= j <= i.
Recurrence:
For P_h, element a is given by: a_(1, 1) = 1; a_(i, j) = ((i+h-2)/(i-1)) (a_(i-1, j) + a_(i-1, j-1)).

Keyword tabf by Michel Marcus, Oct 22 2017

A109495 Entries in 3-dimensional solid related to Prouhet-Tarry problem.

Original entry on oeis.org

1, 2, 2, 1, 2, 1, 1, 1, 2, 5, 2, 1, 5, 5, 1, 1, 2, 1, 1, 2, 1, 2, 8, 8, 2, 1, 8, 15, 8, 1, 2, 8, 8, 2, 1, 2, 1, 1, 3, 3, 1, 2, 11, 18, 11, 2, 1, 11, 31, 31, 11, 1, 3, 18, 31, 18, 3, 3, 11, 11, 3, 1, 2, 1, 1, 4, 6, 4, 1, 2, 14, 32, 32, 14, 2, 1, 14, 53, 80, 53, 14, 1, 4, 32, 80, 80, 32, 4, 6

Offset: 0

Philippe Deléham, Aug 29 2005

Entries of slices [2,k] in A109672, read by rows.

Slice [n,0] gives A046816, slice [0,k] gives A109649, slice [n,n] gives A109673, slice [n,1] gives A109390, slice [1,k] gives A109393.

			Slice [2,0]:
.... 1 ....
... 2 2 ...
.. 1 2 1 ...
Slice [2,1]:
.... 1 1 ....
... 2 5 2 ...
.. 1 5 5 1 ...
... 1 2 1 ...
Slice [2,2]:
.... 1 2 1 ....
... 2 8 8 2 ...
.. 1 8 15 8 1 ...
... 2 8 8 2 ...
.... 1 2 1 ....
Slice [2,3]:
.... 1 3 3 1 ....
... 2 11 18 11 2 ...
.. 1 11 31 31 11 1 ...
... 3 18 31 18 3 ....
.... 3 11 11 3 .....
..... 1 2 1 ......
Slice [2,4]:
.... 1 4 6 4 1 ...
... 2 14 32 32 14 2 ...
.. 1 14 53 80 53 14 1 ...
... 4 32 80 80 32 4 ....
.... 6 32 53 32 6 .....
..... 4 14 14 4 .....;
...... 1 2 1 ......;
Slice [2,5]:
.... 1 5 10 10 5 1 ...
... 2 17 50 70 50 17 2 ...
.. 1 17 81 165 165 81 17 1 ...
... 5 50 165 240 165 50 5 ....
.... 10 70 165 165 70 10 .....
..... 10 50 81 50 10 ......
...... 5 17 17 5 ......
....... 1 2 1 .......

Cf. A109672, A046816, A109649, A109673, A109390, A109393.

Sum of terms in 2D slice [2, k] is 3^(2+k).

A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3k} (n! binomial(n - 3k, j)) / (k!(n - 3k)!3^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 2, 1, 4, 6, 4, 1, 8, 8, 1, 5, 10, 10, 5, 1, 20, 40, 20, 1, 6, 15, 20, 15, 6, 1, 40, 120, 120, 40, 40, 1, 7, 21, 35, 35, 21, 7, 1, 70, 280, 420, 280, 70, 280, 280, 1, 8, 28, 56, 70, 56, 28, 8, 1, 112, 560, 1120, 1120, 560, 112, 1120, 2240, 1120

Offset: 0

Peter Luschny, Jun 04 2021

Consider a sequence of Pascal tetrahedrons (depending on a parameter m >= 1), where the slices of the pyramid are scaled. They are given by the e.g.f.s exp(t^m / m) * exp(t*(x + y)), which provide a sequence of bivariate polynomials in x and y, whose monomials are to be ordered in degree-lexicographic order. For m = 1 one gets A109649 (resp. A046816), for m = 2 one gets A344911 (resp. A344678), and for m = 3 the current triangle. The row sums have an unexpected interpretation in A336614 (see the link).

			Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 2,  1;
[3] 1, 3,  3,  1,  2;
[4] 1, 4,  6,  4,  1,  8,  8;
[5] 1, 5, 10, 10,  5,  1, 20, 40,  20;
[6] 1, 6, 15, 20, 15,  6,  1, 40, 120, 120,  40,  40;
[7] 1, 7, 21, 35, 35, 21,  7,  1,  70, 280, 420, 280, 70, 280, 280.
.
p_{6}(x, y) = x^6 + 6*x^5*y + 15*x^4*y^2 + 20*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + y^6 + 40*x^3 + 120*x^2*y + 120*x*y^2 + 40*y^3 + 40.

mjqxxxx, Proof of conjectured formulas for A336614, Mathematics Stack Exchange.

m=1: A109649, (A046816) [row sums A000244], scaling A007318 [row sums A000079].
m=2: A344911, (A344678) [row sums A005425], scaling A100861 [row sums A000085].
m=3: this triangle      [row sums A336614], scaling A118931 [row sums A001470].

B := (n, k) -> n!/(k!*(n - 3*k)!*(3^k)): C := n -> seq(binomial(n, j), j=0..n):
T := (n, k) -> B(n, k)*C(n - 3*k): seq(seq(T(n, k), k = 0..n/3), n = 0..8);

gf := Exp[t^3 / 3] Exp[t (x + y)]; ser := Series[gf, {t, 0, 9}];
P[n_] := Expand[n! Coefficient[ser, t, n]];
DegLexList[p_] := MonomialList[p, {x, y}, "DegreeLexicographic"] /. x->1 /. y->1;
Table[DegLexList[P[n]], {n, 0, 7}] // Flatten

A109494 Entries in 3-dimensional solid related to Prouhet-Tarry problem.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 5, 5, 1, 2, 5, 2, 1, 1, 1, 2, 1, 2, 8, 8, 1, 1, 8, 15, 8, 1, 2, 8, 8, 2, 1, 2, 1, 1, 2, 1, 3, 11, 11, 3, 3, 18, 31, 18, 3, 1, 11, 31, 31, 11, 1, 2, 11, 18, 11, 2, 1, 3, 3, 1, 1, 2, 1, 4, 14, 14, 4, 6, 32, 53, 32, 6, 4, 32, 80, 80, 32, 4, 1, 14, 53, 80, 53, 14, 1

Offset: 0

Philippe Deléham, Aug 29 2005

Entries of slices [n,2] in A109672, read by rows.

Slice [n,0] gives A046816, slice [0,k] gives A109649, slice [n,n] gives A109673, slice [n,1] gives A109390, slice [1,k] gives A109393, slice [2,k] gives A109495.

			Slice [0,2]:
... 1 2 1 ...
.... 2 2 ....
..... 1 .....
Slice [1,2]:
... 1 2 1 ...
.. 1 5 5 1 ...
... 2 5 2 ...
.... 1 1 ....
Slice [2,2]:
.... 1 2 1 ....
... 2 8 8 2 ...
.. 1 8 15 8 1 ...
... 2 8 8 2 ...
.... 1 2 1 ....
Slice [3,2]:
..... 1 2 1 .....
.... 3 11 11 3 ....
... 3 18 31 18 3 ...
.. 1 11 31 31 11 1 ...
... 2 11 18 11 2 ...
.... 1 3 3 1 ....
Slice [4,2]:
...... 1 2 1 ......
..... 4 14 14 4 .....
.... 6 32 53 32 6 ....
... 4 32 80 80 32 4 ...
.. 1 14 53 80 53 14 1 ...
... 2 14 32 32 14 2 ...
.... 1 4 6 4 1 ....
Slice [5,2]:
....... 1 2 1 .......
...... 5 17 17 5 ......
..... 10 50 81 50 10 .....
.... 10 70 165 165 70 10 ....
... 5 50 165 240 165 50 5 ...
.. 1 17 81 165 165 81 17 1 ...
... 2 17 50 70 50 17 2 ...
.... 1 5 10 10 5 1 ....

Sum of terms in 2D slice [n, 2] is 3^(n+2).

A130046 Hexagonal pyramid of Pascal numbers in 3 dimensions. The 3-dimensional sequence is split into slices of the pyramid which in turn consist of rows of the slice, each containing multiple columns of numbers and where each element of slice j is composed of the sum of the three elements above it in slice j-1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 5, 2, 1, 5, 5, 1, 1, 2, 1, 1, 2, 1, 2, 8, 8, 2, 1, 8, 15, 8, 1, 2, 8, 8, 2, 1, 2, 1, 1, 2, 1, 3, 11, 11, 3, 3, 18, 31, 18, 3, 1, 11, 31, 31, 11, 1, 2, 11, 18, 11, 2, 1, 3, 3, 1, 1, 3, 3, 1, 3, 15, 24, 15, 3, 3, 24, 60, 60, 24, 3, 1, 15, 60, 93, 60, 15, 1, 3, 24

Offset: 0

Jeffrey C. Jacobs (darklord(AT)timehorse.com), May 03 2007

nonn

Successive slices [0,0], [1,0], [1,1], [2,1], [2,2], [3,2], [3,3], [4,3], [4,4], ... in table A109672; see also A046816 (slices [n,0]), A109673 (slices [n,n]), A109649 (slices [0,k]), A109390 (slices [n,1]), A109393 (slices [1,k]), A109494 (slices [n,2]), A109495 (slices [2,k]). - Philippe Deléham, May 03 2007

			Slice[0]:
...
Slice[1]:
1
Slice[2]:
.1
1.1
Slice[3]:
.1.1
1.3.1
.1.1
Slice[4]:
..1.1
.2.5.2
1.5.5.1
.1.2.1
Slice[5]:
....1..2..1
..2..8..8..2
.1..8.15..8..1
..2..8..8..2
....1..2..1
Slice[6]:
.....1..2..1
....3.11.11..3
..3.18.31.18..3
.1.11.31.31.11..1
..2.11.18.11..2
....1..3..3..1

Jeffrey C. Jacobs, Table of n, a(n) for n = 0..30324
Jeffrey C. Jacobs, Python program for this sequence

Cf. A077043, A019298.
Subsequence of A109672 table of slices.
Other tables of slices (see 2007 comment from Philippe Deléham): A046816, A109390, A109393, A109494, A109495, A109649, A109673.
Cf. A007318 (Pascal's triangle).

Let j be a given slice of the hexagonal pyramid. For j = 0, there are no elements.
For j > 0, let a[x] to a[x+y-1] represent the elements of the slice, where x is the (j-1)th element of A019298 and y is the j-th element of A077043. Each slice j consists of j rows of varying column length, numbered 0 to j-1.
The length of the first row of slice j is given by floor((j+1)/2) and the last row by floor(j/2)+1, where by convention the last row is always greater or equal in length to the first row.
The floor(j/2)th row is j columns in length and any row before it is given by the formula floor((j+1)/2) + row#. For rows after the floor(j/2)th row, the length is given by floor(j/2) + j - row#.
The elements a[x] to a[x+y-1] are thus layed out as a concatenated series of rows of varying column lengths as specified above.
Thus for a given slice j, the element at row row# and column col# is represented by a[x + floor((j+1)/2) * row# + row# * (row# - 1) / 2 + col# ] when row# <= floor(j/2) and by a[x + y - (floor(j/2) + 1) * (j - row#) - (j - row#) * (j - row# + 1) / 2 + col#] otherwise, where x and y are defined above and row# and col# start counting from 0.
The elements of a for a given slice j, row# and col#, represented by the coordinate pair (row#, col#), is given by the following recursive relation:
For j = 1, there is 1 element whose value is 1 at (0, 0). Call this Slice[1] whose first and only element forms a0 = 1.
For j > 1, each element (row#, col#) is given by the sum of the 3 elements above it in the pyramid. If the preceding slice does not contain one of the cells specified because the coordinates are invalid for that slice, the value is assumed to be 0.
The cells above can be found using the following formula for a given cell Slice[j](row#, col#):
If j is odd:
If row# > floor(j/2):
Sum:
Slice[j-1](row#, col#-1)
Slice[j-1](row#-1,col#)
Slice[j-1](row#-1,col#-1)
Otherwise:
Sum:
Slice[j-1](row#, col#)
Slice[j-1](row#-1,col#)
Slice[j-1](row#-1,col#-1)
Otherwise:
If row# > floor(j/2):
Sum:
Slice[j-1](row#, col#)
Slice[j-1](row#-1,col#)
Slice[j-1](row#,col#-1)
Otherwise:
Sum:
Slice[j-1](row#, col#)
Slice[j-1](row#,col#-1)
Slice[j-1](row#-1,col#-1)
Each slice is also a solution to the Prouhet-Tarry-Escott problem for a given n and k. The slices[n,k] in sequence A109672 map to the slices here by the relation k + n = j - 1, where k = n (j odd) or k = n + 1 (j even).
When j is even, k = n - 1 would also be a solution to the Pascal hexagonal pyramid, however here the k = n + 1 solution is chosen. For j even, the slices are also given by A109673.
Only 3 of the 6 hexagonal vertices have corresponding cells in the slice below them. Only every other vertex has a cell below it and all vertices with cells below them are always separated by 2 edges.
By convention, when constructing Slice[j] for j odd, the uppermost vertices of Slice[j-1] are chosen to have cells below them and for j even the 2 vertices adjacent to the uppermost vertices of Slice[j-1] are chosen.

cp's OEIS Frontend

A109672 Entries in 3-dimensional solids related to Prouhet-Tarry problem.

Views

Author

Keywords

Comments

Examples

Formula

A109390 Entries in 3-dimensional solid related to Prouhet-Tarry problem.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A109393 Entries in 3-dimensional solid related to Prouhet-Tarry problem.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A178819 Pascal's prism (3-dimensional array) read by folded antidiagonal cross-sections: (h+i; h, i-j, j), h >= 0, i >= 0, 0 <= j <= i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A109495 Entries in 3-dimensional solid related to Prouhet-Tarry problem.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3*k} (n! * binomial(n - 3*k, j)) / (k!*(n - 3*k)!*3^k).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A109494 Entries in 3-dimensional solid related to Prouhet-Tarry problem.

Views

Author

Keywords

Comments

Examples

Formula

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3k} (n! binomial(n - 3k, j)) / (k!(n - 3k)!3^k).