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%I #21 Feb 16 2023 16:20:13
%S 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 -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 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 Array, binomial(j-i,j), read by rising antidiagonals.
%C Inverse of A052509, or A004070???
%H G. C. Greubel, <a href="/A164925/b164925.txt">Antidiagonals n = 0..50, flattened</a>
%F Sum_{k=0..n} T(n, k) = A164965(n). - _Mark Dols_, Sep 02 2009
%F From _G. C. Greubel_, Feb 10 2023: (Start)
%F A(n, k) = binomial(k-n, k), with A(0, k) = A(n, 0) = 1 (array).
%F T(n, k) = binomial(2*k-n, k), with T(n, 0) = T(n, n) = 1 (antidiagonal triangle).
%F Sum_{k=0..n} (-1)^k*T(n, k) = A008346(n).
%F Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^n*A052992(n). (End)
%e Array, A(n, k), begins as:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, -1, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, -2, 1, 0, 0, 0, 0, 0, 0, ...
%e 1, -3, 3, -1, 0, 0, 0, 0, 0, ...
%e 1, -4, 6, -4, 1, 0, 0, 0, 0, ...
%e 1, -5, 10, -10, 5, -1, 0, 0, 0, ...
%e 1, -6, 15, -20, 15, -6, 1, 0, 0, ...
%e 1, -7, 21, -35, 35, -21, 7, -1, 0, ...
%e Antidiagonal triangle, T(n, k), begins as:
%e 1;
%e 1, 1;
%e 1, 0, 1;
%e 1, -1, 0, 1;
%e 1, -2, 0, 0, 1;
%e 1, -3, 1, 0, 0, 1;
%e 1, -4, 3, 0, 0, 0, 1;
%e 1, -5, 6, -1, 0, 0, 0, 1;
%e 1, -6, 10, -4, 0, 0, 0, 0, 1;
%t T[n_, k_]:= If[k==0 || k==n, 1, Binomial[2*k-n, k]];
%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 10 2023 *)
%o (PARI) {A(i, j) = if( i<0, 0, if(i==0 || j==0, 1, binomial(j-i, j)))}; /* _Michael Somos_, Jan 25 2012 */
%o (Magma)
%o A164925:= func< n,k | k eq 0 or k eq n select 1 else Binomial(2*k-n,k) >;
%o [A164925(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Feb 10 2023
%o (SageMath)
%o def A164925(n,k): return 1 if (k==0 or k==n) else binomial(2*k-n, k)
%o flatten([[A164925(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 10 2023
%Y Cf. A004070, A008346, A010892, A052509, A052992.
%Y Cf. A109466, A130595, A164899, A164915, A164965.
%K sign,easy,tabl
%O 0,12
%A _Mark Dols_, Aug 31 2009
%E Edited by _Michael Somos_, Jan 26 2012
%E Offset changed by _G. C. Greubel_, Feb 10 2023
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