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A122432
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Riordan array (1/(1+x)^3,x).
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10
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1, -3, 1, 6, -3, 1, -10, 6, -3, 1, 15, -10, 6, -3, 1, -21, 15, -10, 6, -3, 1, 28, -21, 15, -10, 6, -3, 1, -36, 28, -21, 15, -10, 6, -3, 1, 45, -36, 28, -21, 15, -10, 6, -3, 1, -55, 45, -36, 28, -21, 15, -10
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OFFSET
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0,2
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COMMENTS
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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).
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
/I_k 0\
\ 0 M/
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
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.
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)
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LINKS
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FORMULA
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Number triangle T(n, k) = [k<=n]*(-1)^(n-k)*binomial(n-k+2, 2).
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
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EXAMPLE
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The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 ...
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0: 1
1 :-3 1
2: 6 -3 1
3: -10 6 -3 1
4: 15 -10 6 -3 1
5; -21 15 -10 6 -3 1
6: 28 -21 15 -10 6 -3 1
7: -36 28 -21 15 -10 6 -3 1
8: 45 -36 28 -21 15 -10 6 -3 1
9: -55 45 -36 28 -21 15 -10 6 -3 1
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MATHEMATICA
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Table[(-1)^(n - k)*Binomial[n - k + 2, 2], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 29 2017 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1((-1)^(n-k)*binomial(n-k+2, 2), ", "))) \\ G. C. Greubel, Oct 29 2017
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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