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

%I A129536 #14 May 09 2025 23:12:17
%S A129536 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 A129536 9,45
%N A129536 Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal).
%C A129536 Sequence arrangement:
%C A129536 ....................C(10,2)*C(0,0)
%C A129536 .............C(9,1)*C(1,1)...C(9,2)*C(1,0)
%C A129536 .....C(8,0)*C(2,2)...C(8,1)*C(2,1)...C(8,2)*C(2,0)
%C A129536 ..............C(7,0)*C(3,2)...C(7,1)*C(3,1)...C(7,2)*C(3,0)
%C A129536 .....................C(6,0)*C(4,2)...C(6,1)*C(4,1)...C(6,2)*C(4,0)
%C A129536 .............................C(5,0)*C(5,2)...C(5,1)*C(5,1)...C(5,2)*C(5,0)
%C A129536 .....................................C(4,0)*C(6,2)...C(4,1)*C(6,1)...C(4,2)*C(6,0)
%C A129536 .............................................C(3,0)*C(7,2)...C(3,1)*C(7,1)...C(3,2)*C(7,0)
%C A129536 .....................................................C(2,0)*C(8,2)...C(2,1)*C(8,1)...C(2,2)*C(8,0)
%C A129536 .............................................................C(1,0)*C(9,2)...C(1,1)*C(9,1)
%C A129536 .....................................................................C(0,0)*C(10,2)
%C A129536 "m" matching: analog (permutations with exactly "m" fixed points.
%C A129536 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 A129536 if compared bbbbbbbbbb (b 10 times letters then 45 * "8" matching.(sum 45)
%C A129536 ("8" matching: analog (permutations with exactly 8 fixed points.).
%C A129536 If aabbbbbbbb (a 2 letters b 8 letters) permutations compared
%C A129536 aabbbbbbbb (a twice letters b 8 times letters)then 1 * "10"
%C A129536 matching),16 * "8" matching, 28 * "6" matching (sum 45)
%C A129536 If aabbbbbbbb (a 8 letters b 2 letters)permutations compared
%C A129536 aaaaaaaabb (a 8 times letters b twice letters)then 1 * "0"
%C A129536 matching),16 * "2" matching, 28 * "4" matching (sum 45)
%C A129536 all rows (sum 45)
%C A129536 etc...
%C A129536 matching equialent or analog "fixed points"
%C A129536 example:
%C A129536 arrangement relevant!
%C A129536 compared
%C A129536 letters..
%C A129536 times....
%C A129536 a...b
%C A129536 matching:..........0..1..2..3..4..5..6..7..8..9..10 0..10...................45.........................
%C A129536 .1..9.................9....36......................
%C A129536 .2..8..............1.....16....28..................
%C A129536 .3..7.................3....21.....21................
%C A129536 .4..6....................6....24....15............
%C A129536 .5..5......................10....25.....10.........
%C A129536 .6..4.........................15....24.....6........
%C A129536 .7..3.............................21....21.....3....
%C A129536 .8..2................................28....16......1
%C A129536 .9..1...................................36.....9....
%C A129536 10..0......................................45.......
%C A129536 matching:...........0..1..2..3..4..5..6..7..8..9..10
%C A129536 The Maple code produces this:
%C A129536 45, 36, 28, 21, 15, 10, 6, 3, 1
%C A129536 9, 16, 21, 24, 25, 24, 21, 16, 9
%C A129536 1, 3, 6, 10, 15, 21, 28, 36, 45
%C A129536 This is the table rotated right by Pi/4.
%H A129536 Johnph77, <a href="https://web.archive.org/web/20090627142143/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)
%p A129536 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 A129536 Cf. A113899.
%K A129536 fini,nonn,uned
%O A129536 0,1
%A A129536 _Zerinvary Lajos_, May 29 2007
%E A129536 Edited by _Charles R Greathouse IV_, Oct 28 2009