This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A164925 #21 Feb 16 2023 16:20:13
%S A164925 1,1,1,1,0,1,1,-1,0,1,1,-2,0,0,1,1,-3,1,0,0,1,1,-4,3,0,0,0,1,1,-5,6,
%T A164925 -1,0,0,0,1,1,-6,10,-4,0,0,0,0,1,1,-7,15,-10,1,0,0,0,0,1,1,-8,21,-20,
%U A164925 5,0,0,0,0,0,1,1,-9,28,-35,15,-1,0,0,0,0,0,1,1,-10,36,-56,35,-6,0,0,0,0,0,0,1
%N A164925 Array, binomial(j-i,j), read by rising antidiagonals.
%C A164925 Inverse of A052509, or A004070???
%H A164925 G. C. Greubel, <a href="/A164925/b164925.txt">Antidiagonals n = 0..50, flattened</a>
%F A164925 Sum_{k=0..n} T(n, k) = A164965(n). - _Mark Dols_, Sep 02 2009
%F A164925 From _G. C. Greubel_, Feb 10 2023: (Start)
%F A164925 A(n, k) = binomial(k-n, k), with A(0, k) = A(n, 0) = 1 (array).
%F A164925 T(n, k) = binomial(2*k-n, k), with T(n, 0) = T(n, n) = 1 (antidiagonal triangle).
%F A164925 Sum_{k=0..n} (-1)^k*T(n, k) = A008346(n).
%F A164925 Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^n*A052992(n). (End)
%e A164925 Array, A(n, k), begins as:
%e A164925 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A164925 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A164925 1, -1, 0, 0, 0, 0, 0, 0, 0, ...
%e A164925 1, -2, 1, 0, 0, 0, 0, 0, 0, ...
%e A164925 1, -3, 3, -1, 0, 0, 0, 0, 0, ...
%e A164925 1, -4, 6, -4, 1, 0, 0, 0, 0, ...
%e A164925 1, -5, 10, -10, 5, -1, 0, 0, 0, ...
%e A164925 1, -6, 15, -20, 15, -6, 1, 0, 0, ...
%e A164925 1, -7, 21, -35, 35, -21, 7, -1, 0, ...
%e A164925 Antidiagonal triangle, T(n, k), begins as:
%e A164925 1;
%e A164925 1, 1;
%e A164925 1, 0, 1;
%e A164925 1, -1, 0, 1;
%e A164925 1, -2, 0, 0, 1;
%e A164925 1, -3, 1, 0, 0, 1;
%e A164925 1, -4, 3, 0, 0, 0, 1;
%e A164925 1, -5, 6, -1, 0, 0, 0, 1;
%e A164925 1, -6, 10, -4, 0, 0, 0, 0, 1;
%t A164925 T[n_, k_]:= If[k==0 || k==n, 1, Binomial[2*k-n, k]];
%t A164925 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 10 2023 *)
%o A164925 (PARI) {A(i, j) = if( i<0, 0, if(i==0 || j==0, 1, binomial(j-i, j)))}; /* _Michael Somos_, Jan 25 2012 */
%o A164925 (Magma)
%o A164925 A164925:= func< n,k | k eq 0 or k eq n select 1 else Binomial(2*k-n,k) >;
%o A164925 [A164925(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 10 2023
%o A164925 (SageMath)
%o A164925 def A164925(n,k): return 1 if (k==0 or k==n) else binomial(2*k-n, k)
%o A164925 flatten([[A164925(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 10 2023
%Y A164925 Cf. A004070, A008346, A010892, A052509, A052992.
%Y A164925 Cf. A109466, A130595, A164899, A164915, A164965.
%K A164925 sign,easy,tabl
%O A164925 0,12
%A A164925 _Mark Dols_, Aug 31 2009
%E A164925 Edited by _Michael Somos_, Jan 26 2012
%E A164925 Offset changed by _G. C. Greubel_, Feb 10 2023