A113899 -id:A113899 - cp's OEIS frontend

A129352 Number parallelogram based on Pascal's triangle, read along diagonals.

70, 35, 35, 15, 40, 15, 5, 30, 30, 5, 1, 16, 36, 16, 1, 5, 30, 30, 5, 15, 40, 15, 35, 35, 70

Offset: 0

Zerinvary Lajos, May 28 2007

			Terms are the following square, read by diagonals:
  70, 35, 15,  5,  1
  35, 40, 30, 16,  5
  15, 30, 36, 30, 15
   5, 16, 30, 40, 35
   1,  5, 15, 35, 70

Cf. A113899.

with(combinat):T:=(n,i)->binomial(i,n)*binomial(8-i,4-n): for n from 0 to 4 do seq(T(n, i), i=0+n..8-4+n) od;

Edited slightly by Charles R Greathouse IV, Oct 28 2009

A129536 Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal).

Original entry on oeis.org

45, 9, 36, 1, 16, 28, 3, 21, 21, 6, 24, 15, 10, 25, 10, 15, 24, 6, 21, 21, 3, 28, 16, 1, 36, 9, 45

Offset: 0

Zerinvary Lajos, May 29 2007

Sequence arrangement:
....................C(10,2)*C(0,0)
.............C(9,1)*C(1,1)...C(9,2)*C(1,0)
.....C(8,0)*C(2,2)...C(8,1)*C(2,1)...C(8,2)*C(2,0)
..............C(7,0)*C(3,2)...C(7,1)*C(3,1)...C(7,2)*C(3,0)
.....................C(6,0)*C(4,2)...C(6,1)*C(4,1)...C(6,2)*C(4,0)
.............................C(5,0)*C(5,2)...C(5,1)*C(5,1)...C(5,2)*C(5,0)
.....................................C(4,0)*C(6,2)...C(4,1)*C(6,1)...C(4,2)*C(6,0)
.............................................C(3,0)*C(7,2)...C(3,1)*C(7,1)...C(3,2)*C(7,0)
.....................................................C(2,0)*C(8,2)...C(2,1)*C(8,1)...C(2,2)*C(8,0)
.............................................................C(1,0)*C(9,2)...C(1,1)*C(9,1)
.....................................................................C(0,0)*C(10,2)
"m" matching: analog (permutations with exactly "m" fixed points.
if aabbbbbbbb (a twice letters b 8 times letters) permutations compared aaaaaaaaaa (a 10 times letters) then 45 * "2" matching.(sum 45)("2" matching: analog(permutations with exactly 2 fixed points.)
if compared bbbbbbbbbb (b 10 times letters then 45 * "8" matching.(sum 45)
("8" matching: analog (permutations with exactly 8 fixed points.).
If aabbbbbbbb (a 2 letters b 8 letters) permutations compared
aabbbbbbbb (a twice letters b 8 times letters)then 1 * "10"
matching),16 * "8" matching, 28 * "6" matching (sum 45)
If aabbbbbbbb (a 8 letters b 2 letters)permutations compared
aaaaaaaabb (a 8 times letters b twice letters)then 1 * "0"
matching),16 * "2" matching, 28 * "4" matching (sum 45)
all rows (sum 45)
etc...
matching equialent or analog "fixed points"
example:
arrangement relevant!
compared
letters..
times....
a...b
matching:..........0..1..2..3..4..5..6..7..8..9..10 0..10...................45.........................
.1..9.................9....36......................
.2..8..............1.....16....28..................
.3..7.................3....21.....21................
.4..6....................6....24....15............
.5..5......................10....25.....10.........
.6..4.........................15....24.....6........
.7..3.............................21....21.....3....
.8..2................................28....16......1
.9..1...................................36.....9....
10..0......................................45.......
matching:...........0..1..2..3..4..5..6..7..8..9..10
The Maple code produces this:
45, 36, 28, 21, 15, 10, 6, 3, 1
9, 16, 21, 24, 25, 24, 21, 16, 9
1, 3, 6, 10, 15, 21, 28, 36, 45
This is the table rotated right by Pi/4.

Johnph77, Lottery Numeric Positional Frequency Charts Note:Information herein is intended for lottery system developers, analysts and operators. It is not intended for gaming purposes. 3/11 table: (Horizontal > Total: 165, Vertical > Total: 45)

Cf. A113899.

with(combinat):T:=(n,i)->binomial(i,n)*binomial(10-i,2-n): for n from 0 to 2 do seq(T(n, i), i=0+n..10-2+n) od;

Edited by Charles R Greathouse IV, Oct 28 2009

cp's OEIS Frontend

A129352 Number parallelogram based on Pascal's triangle, read along diagonals.

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