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A014473
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Pascal's triangle - 1.
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14
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0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 9, 9, 4, 0, 0, 5, 14, 19, 14, 5, 0, 0, 6, 20, 34, 34, 20, 6, 0, 0, 7, 27, 55, 69, 55, 27, 7, 0, 0, 8, 35, 83, 125, 125, 83, 35, 8, 0, 0, 9, 44, 119, 209, 251, 209, 119, 44, 9, 0, 0, 10, 54, 164, 329, 461, 461, 329, 164, 54, 10, 0, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,8
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COMMENTS
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Indexed as a square array A(n,k): If X is an (n+k)-set and Y a fixed k-subset of X then A(n,k) is equal to the number of n-subsets of X intersecting Y. - Peter Luschny, Apr 20 2012
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LINKS
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FORMULA
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G.f.: x^2*y/((1 - x)*(1 - x*y)*(1 - x*(1 + y))). - Ralf Stephan, Jan 24 2005
T(n, k) = T(n-1, k-1) + T(n-1, k) + 1, 0 < k < n with T(n, 0) = T(n, n) = 0. - Reinhard Zumkeller, Jul 18 2015
If seen as a square array read by antidiagonals the generating function of row n is: G(n) = (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1)). - Peter Luschny, Feb 13 2019
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EXAMPLE
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Triangle begins:
0;
0, 0;
0, 1, 0;
0, 2, 2, 0;
0, 3, 5, 3, 0;
0, 4, 9, 9, 4, 0;
0, 5, 14, 19, 14, 5, 0;
0, 6, 20, 34, 34, 20, 6, 0;
...
Seen as a square array read by antidiagonals:
[0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000004
[1] 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... A001477
[2] 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, ... A000096
[3] 0, 3, 9, 19, 34, 55, 83, 119, 164, 219, 285, 363, ... A062748
[4] 0, 4, 14, 34, 69, 125, 209, 329, 494, 714, 1000, 1364, ... A063258
[5] 0, 5, 20, 55, 125, 251, 461, 791, 1286, 2001, 3002, 4367, ... A062988
[6] 0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, ... A124089
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MAPLE
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with(combstruct): for n from 0 to 11 do seq(-1+count(Combination(n), size=m), m = 0 .. n) od; # Zerinvary Lajos, Apr 09 2008
# The rows of the square array:
Arow := proc(n, len) local gf, ser;
gf := (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1));
ser := series(gf, x, len+2): seq((-1)^(n+1)*coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do lprint([n], Arow(n, 12)) od; # Peter Luschny, Feb 13 2019
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PROG
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(Haskell)
a014473 n k = a014473_tabl !! n !! k
a014473_row n = a014473_tabl !! n
a014473_tabl = map (map (subtract 1)) a007318_tabl
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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