A106509 - OEIS

%I #26 Jul 02 2021 16:51:12

%S 1,0,1,-1,-1,1,1,0,-2,1,0,1,2,-3,1,-1,-1,-1,5,-4,1,1,0,0,-6,9,-5,1,0,

%T 1,0,6,-15,14,-6,1,-1,-1,1,-6,21,-29,20,-7,1,1,0,-2,7,-27,50,-49,27,

%U -8,1,0,1,2,-9,34,-77,99,-76,35,-9,1,-1,-1,-1,11,-43,111,-176,175,-111,44,-10,1

%N Riordan array ((1+x)/(1+x+x^2), x/(1+x)), read by rows.

%C Row sums are A106510.

%C Diagonal sums are A106511.

%C Inverse of A072405 (when this starts 1, 0, 1, ...).

%H G. C. Greubel, <a href="/A106509/b106509.txt">Rows n = 0..50 of the triangle, flattened</a>

%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.

%F T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(2n-k-j, j).

%F T(n,k) = T(n-1,k-1) - 2*T(n-1,k) + T(n-2,k-1) - 2*T(n-2,k) + T(n-3,k-1) - T(n-3,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = 0, T(2,1) = T(2,0) = -1, T(n,k) = 0 if k<0 or if k>n. - _Philippe Deléham_, Jan 12 2014

%F Sum_{k=0..n} T(n,k) = A106510(n). - _G. C. Greubel_, Apr 28 2021

%e Triangle begins:

%e    1;

%e    0,  1;

%e   -1, -1,  1;

%e    1,  0, -2,  1;

%e    0,  1,  2, -3,  1;

%e   -1, -1, -1,  5, -4,  1;

%t (* The function RiordanArray is defined in A256893. *)

%t RiordanArray[(1 + #)/(1 + # + #^2)&, #/(1 + #)&, 12] // Flatten (* _Jean-François Alcover_, Jul 19 2019 *)

%o (Magma)

%o T:= func< n,k | (&+[ (-1)^j*Binomial(2*n-k-j, j): j in [0..n-k]]) >;

%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 28 2021

%o (Sage)

%o def T(n,k): return sum( (-1)^j*binomial(2*n-k-j, j) for j in (0..n-k))

%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 28 2021

%Y Cf. A072405, A106510, A106511.

%K easy,sign,tabl

%O 0,9

%A _Paul Barry_, May 04 2005