A164925 Array, binomial(j-i,j), read by rising antidiagonals.
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, -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, 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
Offset: 0
Examples
Array, A(n, k), begins as: 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 0, 0, 0, 0, 0, 0, 0, 0, ... 1, -1, 0, 0, 0, 0, 0, 0, 0, ... 1, -2, 1, 0, 0, 0, 0, 0, 0, ... 1, -3, 3, -1, 0, 0, 0, 0, 0, ... 1, -4, 6, -4, 1, 0, 0, 0, 0, ... 1, -5, 10, -10, 5, -1, 0, 0, 0, ... 1, -6, 15, -20, 15, -6, 1, 0, 0, ... 1, -7, 21, -35, 35, -21, 7, -1, 0, ... Antidiagonal triangle, T(n, k), begins as: 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, -1, 0, 0, 0, 1; 1, -6, 10, -4, 0, 0, 0, 0, 1;
Links
- G. C. Greubel, Antidiagonals n = 0..50, flattened
Crossrefs
Programs
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Magma
A164925:= func< n,k | k eq 0 or k eq n select 1 else Binomial(2*k-n,k) >; [A164925(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 10 2023
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Mathematica
T[n_, k_]:= If[k==0 || k==n, 1, Binomial[2*k-n, k]]; Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 10 2023 *) -
PARI
{A(i, j) = if( i<0, 0, if(i==0 || j==0, 1, binomial(j-i, j)))}; /* Michael Somos, Jan 25 2012 */ -
SageMath
def A164925(n,k): return 1 if (k==0 or k==n) else binomial(2*k-n, k) flatten([[A164925(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 10 2023
Formula
From G. C. Greubel, Feb 10 2023: (Start)
A(n, k) = binomial(k-n, k), with A(0, k) = A(n, 0) = 1 (array).
T(n, k) = binomial(2*k-n, k), with T(n, 0) = T(n, n) = 1 (antidiagonal triangle).
Sum_{k=0..n} (-1)^k*T(n, k) = A008346(n).
Sum_{k=0..n} (-2)^k*T(n, k) = (-1)^n*A052992(n). (End)
Extensions
Edited by Michael Somos, Jan 26 2012
Offset changed by G. C. Greubel, Feb 10 2023
Comments