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%I #55 Apr 08 2024 09:25:15
%S 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,
%T 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,
%U 119,209,251,209,119,44,9,0,0,10,54,164,329,461,461,329,164,54,10,0
%N Pascal's triangle - 1: Triangle read by rows: T(n, k) = A007318(n, k) - 1.
%C 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
%H Reinhard Zumkeller, <a href="/A014473/b014473.txt">Rows n=0..100 of triangle, flattened</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%F G.f.: x^2*y/((1 - x)*(1 - x*y)*(1 - x*(1 + y))). - _Ralf Stephan_, Jan 24 2005
%F T(n,k) = A109128(n,k) - A007318(n,k), 0 <= k <= n. - _Reinhard Zumkeller_, Apr 10 2012
%F 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
%F 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
%F From _G. C. Greubel_, Apr 08 2024: (Start)
%F T(n, n-k) = T(n, k).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A129696(n-2).
%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = b(n-1), where b(n) is the repeating pattern {0, 0, -1, -2, -1, 1, 1, -1, -2, -1, 0, 0}_{n=0..11}, with b(n) = b(n-12). (End)
%e Triangle begins:
%e 0;
%e 0, 0;
%e 0, 1, 0;
%e 0, 2, 2, 0;
%e 0, 3, 5, 3, 0;
%e 0, 4, 9, 9, 4, 0;
%e 0, 5, 14, 19, 14, 5, 0;
%e 0, 6, 20, 34, 34, 20, 6, 0;
%e ...
%e Seen as a square array read by antidiagonals:
%e [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... A000004
%e [1] 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... A001477
%e [2] 0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, ... A000096
%e [3] 0, 3, 9, 19, 34, 55, 83, 119, 164, 219, 285, 363, ... A062748
%e [4] 0, 4, 14, 34, 69, 125, 209, 329, 494, 714, 1000, 1364, ... A063258
%e [5] 0, 5, 20, 55, 125, 251, 461, 791, 1286, 2001, 3002, 4367, ... A062988
%e [6] 0, 6, 27, 83, 209, 461, 923, 1715, 3002, 5004, 8007, 12375, ... A124089
%p 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
%p # The rows of the square array:
%p Arow := proc(n, len) local gf, ser;
%p gf := (x - 1)^(-n - 1) + (-1)^(n + 1)/(x*(x - 1));
%p ser := series(gf, x, len+2): seq((-1)^(n+1)*coeff(ser, x, j), j=0..len) end:
%p for n from 0 to 9 do lprint([n], Arow(n, 12)) od; # _Peter Luschny_, Feb 13 2019
%t Table[Binomial[n,k] -1, {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 08 2024 *)
%o (Haskell)
%o a014473 n k = a014473_tabl !! n !! k
%o a014473_row n = a014473_tabl !! n
%o a014473_tabl = map (map (subtract 1)) a007318_tabl
%o -- _Reinhard Zumkeller_, Apr 10 2012
%o (Magma)
%o [Binomial(n,k)-1: k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 08 2024
%o (SageMath)
%o flatten([[binomial(n,k)-1 for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 08 2024
%Y Triangle without zeros: A014430.
%Y Related: A323211 (A007318(n, k) + 1).
%Y A000295 (row sums), A059841 (alternating row sums), A030662(n-1) (central terms).
%Y Columns include A000096, A062748, A062988, A063258.
%Y Diagonals of A(n, n+d): A030662 (d=0), A010763 (d=1), A322938 (d=2).
%Y Cf. A000004, A001477, A007318, A030662, A059841, A109128, A124089, A129696.
%K nonn,tabl,easy
%O 0,8
%A _N. J. A. Sloane_
%E More terms from _Erich Friedman_
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