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A127895
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Riordan array (1/(1+x)^3, x/(1+x)^3).
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3
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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
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OFFSET
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0,2
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COMMENTS
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The matrix inverse of the convolution triangle of A001764 (number of ternary trees). - Peter Luschny, Oct 09 2022
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LINKS
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FORMULA
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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).
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EXAMPLE
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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;
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MAPLE
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# Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022
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MATHEMATICA
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Table[(-1)^(n-k)*Binomial[n+2*k+2, n-k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 29 2018 *)
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PROG
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(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
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CROSSREFS
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Alternating sign version of A127893.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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