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Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal).
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%I #7 Jul 11 2015 16:49:37

%S 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,

%T 9,45

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

%C Sequence arrangement:

%C ....................C(10,2)*C(0,0)

%C .............C(9,1)*C(1,1)...C(9,2)*C(1,0)

%C .....C(8,0)*C(2,2)...C(8,1)*C(2,1)...C(8,2)*C(2,0)

%C ..............C(7,0)*C(3,2)...C(7,1)*C(3,1)...C(7,2)*C(3,0)

%C .....................C(6,0)*C(4,2)...C(6,1)*C(4,1)...C(6,2)*C(4,0)

%C .............................C(5,0)*C(5,2)...C(5,1)*C(5,1)...C(5,2)*C(5,0)

%C .....................................C(4,0)*C(6,2)...C(4,1)*C(6,1)...C(4,2)*C(6,0)

%C .............................................C(3,0)*C(7,2)...C(3,1)*C(7,1)...C(3,2)*C(7,0)

%C .....................................................C(2,0)*C(8,2)...C(2,1)*C(8,1)...C(2,2)*C(8,0)

%C .............................................................C(1,0)*C(9,2)...C(1,1)*C(9,1)

%C .....................................................................C(0,0)*C(10,2)

%C "m" matching: analog (permutations with exactly "m" fixed points.

%C 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.)

%C if compared bbbbbbbbbb (b 10 times letters then 45 * "8" matching.(sum 45)

%C ("8" matching: analog (permutations with exactly 8 fixed points.).

%C If aabbbbbbbb (a 2 letters b 8 letters) permutations compared

%C aabbbbbbbb (a twice letters b 8 times letters)then 1 * "10"

%C matching),16 * "8" matching, 28 * "6" matching (sum 45)

%C If aabbbbbbbb (a 8 letters b 2 letters)permutations compared

%C aaaaaaaabb (a 8 times letters b twice letters)then 1 * "0"

%C matching),16 * "2" matching, 28 * "4" matching (sum 45)

%C all rows (sum 45)

%C etc...

%C matching equialent or analog "fixed points"

%C example:

%C arrangement relevant!

%C compared

%C letters..

%C times....

%C a...b

%C matching:..........0..1..2..3..4..5..6..7..8..9..10 0..10...................45.........................

%C .1..9.................9....36......................

%C .2..8..............1.....16....28..................

%C .3..7.................3....21.....21................

%C .4..6....................6....24....15............

%C .5..5......................10....25.....10.........

%C .6..4.........................15....24.....6........

%C .7..3.............................21....21.....3....

%C .8..2................................28....16......1

%C .9..1...................................36.....9....

%C 10..0......................................45.......

%C matching:...........0..1..2..3..4..5..6..7..8..9..10

%C The Maple code produces this:

%C 45, 36, 28, 21, 15, 10, 6, 3, 1

%C 9, 16, 21, 24, 25, 24, 21, 16, 9

%C 1, 3, 6, 10, 15, 21, 28, 36, 45

%C This is the table rotated right by Pi/4.

%H <a href="http://www.johnph77.com/math/lf.html#c4094">Lottery Numeric Positional Frequency Charts</a> 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]

%p 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;

%Y Cf. A113899.

%K fini,nonn

%O 0,1

%A _Zerinvary Lajos_, May 29 2007

%E Edited by _Charles R Greathouse IV_, Oct 28 2009