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A127895 Riordan array (1/(1+x)^3, x/(1+x)^3). 3
1, -3, 1, 6, -6, 1, -10, 21, -9, 1, 15, -56, 45, -12, 1, -21, 126, -165, 78, -15, 1, 28, -252, 495, -364, 120, -18, 1, -36, 462, -1287, 1365, -680, 171, -21, 1, 45, -792, 3003, -4368, 3060, -1140, 231, -24, 1, -55, 1287, -6435, 12376, -11628, 5985, -1771, 300, -27, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Rows n=0..100 of triangle, flattened

FORMULA

T(n, k) = (-1)^(n-k)*binomial(n +2*k +2, n-k).

Sum_{k=0..n} T(n, k) = A127896(n) (row sums).

Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A095263(n) (diagonal sums).

EXAMPLE

Triangle begins

    1;

   -3,     1;

    6,    -6,     1;

  -10,    21,    -9,      1;

   15,   -56,    45,    -12,      1;

  -21,   126,  -165,     78,    -15,      1;

   28,  -252,   495,   -364,    120,    -18,     1;

  -36,   462, -1287,   1365,   -680,    171,   -21,     1;

   45,  -792,  3003,  -4368,   3060,  -1140,   231,   -24,   1;

  -55,  1287, -6435,  12376, -11628,   5985, -1771,   300, -27,   1;

   66, -2002, 12870, -31824,  38760, -26334, 10626, -2600, 378, -30, 1;

MATHEMATICA

Table[(-1)^(n-k)*Binomial[n+2*k+2, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 29 2018 *)

PROG

(PARI) for(n=0, 10, for(k=0, n, print1((-1)^(n-k)*binomial(n+2*k+2, n-k), ", "))) \\ G. C. Greubel, Apr 29 2018

(MAGMA) [(-1)^(n-k)*Binomial(n+2*k+2, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Apr 29 2018

(Sage) flatten([[(-1)^(n-k)*binomial(n+2*k+2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

CROSSREFS

Inverse is A127898.

Alternating sign version of A127893.

Cf. A095263, A127896.

Sequence in context: A124847 A249251 A127893 * A325005 A325013 A152685

Adjacent sequences:  A127892 A127893 A127894 * A127896 A127897 A127898

KEYWORD

easy,sign,tabl,changed

AUTHOR

Paul Barry, Feb 04 2007

EXTENSIONS

Terms a(50) onward added by G. C. Greubel, Apr 29 2018

STATUS

approved

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Last modified April 22 09:10 EDT 2021. Contains 343174 sequences. (Running on oeis4.)