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%I #25 Sep 08 2022 08:45:28
%S 1,-3,1,6,-3,1,-10,6,-3,1,15,-10,6,-3,1,-21,15,-10,6,-3,1,28,-21,15,
%T -10,6,-3,1,-36,28,-21,15,-10,6,-3,1,45,-36,28,-21,15,-10,6,-3,1,-55,
%U 45,-36,28,-21,15,-10
%N Riordan array (1/(1+x)^3,x).
%C Sequence array for (-1)^n*C(n+2,2). Inverse of A122431. Row sums are -A083392(n+1). Antidiagonal sums are (-1)^n*A002623(n).
%C Call the unsigned version of this array M and for k = 0,1,2,... define M(k) to be the lower unit triangular block array
%C /I_k 0\
%C \ 0 M/
%C having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. The infinite matrix product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to A127893. - _Peter Bala_, Jul 22 2014
%C From _Wolfdieter Lang_, Apr 05 2020: (Start)
%C Triangle T(n, k) has the k=0 column (-1)^n*A000217(n+1) = (-1)^n*binomial(n+2, 2), then repeated and down-shifted.
%C The unsigned triangle, i.e., Tup(n, k) := (-1)^(n-k)*T(n-1,k-1) = binomial(n-k+2, 2) with n >= 1, k = 1..n, gives the number of triangles of length k (in some units), for k = 1..n, in the matchstick arrangement (or tower of cards, with n cards as basis) with an enclosing triangle of length n, but only triangles with orientation (up) like the enclosing triangle are counted. The total number of matchsticks (cards) is 3*A000217(n). (See the comment by _Andrew Howroyd_ in A085691). Recurrence: Tup(n, k) = 0 for n < k, Tup(1, 1) = 1, and Tup(n, k) = Tup(n-1, k) + n - k + 1, for n >= 2, k = 1..n. Row sums give A000292(n). (End)
%H G. C. Greubel, <a href="/A122432/b122432.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F Number triangle T(n, k) = [k<=n]*(-1)^(n-k)*binomial(n-k+2, 2).
%F Recurrence: T(n, k) = - T(n-1, k) + (-1)^(n-k)*(n-k+1), for n >= 0, and k = 0..n. - _Wolfdieter Lang_, Apr 06 2020
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 ...
%e -------------------------------------------
%e 0: 1
%e 1 :-3 1
%e 2: 6 -3 1
%e 3: -10 6 -3 1
%e 4: 15 -10 6 -3 1
%e 5; -21 15 -10 6 -3 1
%e 6: 28 -21 15 -10 6 -3 1
%e 7: -36 28 -21 15 -10 6 -3 1
%e 8: 45 -36 28 -21 15 -10 6 -3 1
%e 9: -55 45 -36 28 -21 15 -10 6 -3 1
%e ... reformattet by - _Wolfdieter Lang_, Apr 05 2020
%t Table[(-1)^(n - k)*Binomial[n - k + 2, 2], {n, 0, 49}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Oct 29 2017 *)
%o (PARI) for(n=0,10, for(k=0,n, print1((-1)^(n-k)*binomial(n-k+2,2), ", "))) \\ _G. C. Greubel_, Oct 29 2017
%o (Magma) /* As triangle */ [[(-1)^(n-k)*Binomial(n-k+2, 2): k in [1..n]]: n in [1..10]]; // _G. C. Greubel_, Oct 29 2017
%Y Cf. A000217, A000292, A083392, A085691, A122431, A127893.
%K easy,sign,tabl
%O 0,2
%A _Paul Barry_, Sep 04 2006
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