A122432 Riordan array (1/(1+x)^3,x).

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

Offset: 0

Author

Paul Barry, Sep 04 2006

Keywords

• easy
• sign
• tabl

Comments

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

From Wolfdieter Lang, Apr 05 2020: (Start)

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)

Examples

			The triangle T(n, k) begins:
n\k  0   1   2   3   4   5   6  7  8  9 ...
-------------------------------------------
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
... reformattet by - _Wolfdieter Lang_, Apr 05 2020
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A000217, A000292, A083392, A085691, A122431, A127893.

Programs

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

Mathematica

Table[(-1)^(n - k)*Binomial[n - k + 2, 2], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 29 2017 *)

PARI

for(n=0,10, for(k=0,n, print1((-1)^(n-k)*binomial(n-k+2,2), ", "))) \\ G. C. Greubel, Oct 29 2017

Formula

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