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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.)