A135221 Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.
1, 0, 1, 2, 1, 1, 0, 4, 2, 1, 2, 3, 7, 3, 1, 0, 6, 9, 11, 4, 1, 2, 5, 16, 19, 16, 5, 1, 0, 8, 20, 36, 34, 22, 6, 1, 2, 7, 29, 55, 71, 55, 29, 7, 1, 0, 10, 35, 85, 125, 127, 83, 37, 8, 1, 2, 9, 46, 119, 211, 251, 211, 119, 46, 9, 1, 0, 12, 54, 166, 329, 463, 461, 331, 164, 56, 10, 1
Offset: 0
Examples
First few rows of the triangle: 1; 0, 1; 2, 1, 1; 0, 4, 2, 1; 2, 3, 7, 3, 1; 0, 6, 9, 11, 4, 1; 2, 5, 16, 19, 16, 5, 1; 0, 8, 20, 36, 34, 22, 6, 1; 2, 7, 29, 55, 71, 55, 29, 7, 1; ...
Links
- G. C. Greubel, Rows n = 0..100 of triangle, flattened
Programs
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GAP
T:= function(n,k) if k=n then return 1; else return Binomial(n,k) + (-1)^(n-k); fi; end; Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 20 2019 -
Magma
T:= func< n,k | k eq n select 1 else Binomial(n,k) +(-1)^(n-k) >; [T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019
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Maple
seq(seq( `if`(k=n, 1, binomial(n,k) + (-1)^(n-k)), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019
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Mathematica
T[n_, k_]:= T[n, k]= If[k==n, 1, Binomial[n, k] + (-1)^(n-k)] ; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *) -
PARI
T(n,k) = if(k==n, 1, binomial(n,k) + (-1)^(n-k)); \\ G. C. Greubel, Nov 20 2019
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Sage
def T(n, k): if (k==n): return 1 else: return binomial(n,k) + (-1)^(n-k) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019
Formula
T(n,k) = A007318 + A000012(signed) - Identity matrix, where A000012(signed) = (1; -1,1; 1,-1,1; ...).
T(n,k) = (-1)^(n-k) + binomial(n,k), with T(n,n)=1. - G. C. Greubel, Nov 20 2019
Extensions
More terms added by G. C. Greubel, Nov 20 2019
Comments