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 A122432 Riordan array (1/(1+x)^3,x). 10
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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) LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened 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 EXAMPLE 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 MATHEMATICA Table[(-1)^(n - k)*Binomial[n - k + 2, 2], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 29 2017 *) PROG (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 CROSSREFS Cf. A000217, A000292, A083392, A085691, A122431, A127893. Sequence in context: A145367 A124928 A249250 * A131110 A133093 A065567 Adjacent sequences:  A122429 A122430 A122431 * A122433 A122434 A122435 KEYWORD easy,sign,tabl AUTHOR Paul Barry, Sep 04 2006 STATUS approved

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Last modified September 24 14:49 EDT 2022. Contains 356932 sequences. (Running on oeis4.)