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A129536 Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..26.

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) [From Zerinvary Lajos, Apr 02 2009]

MAPLE

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;

CROSSREFS

Cf. A113899.

Sequence in context: A037218 A070818 A285893 * A108909 A033365 A196093

Adjacent sequences:  A129533 A129534 A129535 * A129537 A129538 A129539

KEYWORD

fini,nonn

AUTHOR

Zerinvary Lajos, May 29 2007

EXTENSIONS

Edited by Charles R Greathouse IV, Oct 28 2009

STATUS

approved

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Last modified January 24 16:47 EST 2020. Contains 331209 sequences. (Running on oeis4.)