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A135221 Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.

%I A135221 #9 Mar 27 2022 18:26:14
%S A135221 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,
%T A135221 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,
%U A135221 46,119,211,251,211,119,46,9,1,0,12,54,166,329,463,461,331,164,56,10,1
%N A135221 Triangle A007318 + A000012(signed) - I, I = Identity matrix, read by rows.
%C A135221 row sums = A051049: (1, 1, 4, 7, 16, 31, 64, ...).
%H A135221 G. C. Greubel, <a href="/A135221/b135221.txt">Rows n = 0..100 of triangle, flattened</a>
%F A135221 T(n,k) = A007318 + A000012(signed) - Identity matrix, where A000012(signed) = (1; -1,1; 1,-1,1; ...).
%F A135221 T(n,k) = (-1)^(n-k) + binomial(n,k), with T(n,n)=1. - _G. C. Greubel_, Nov 20 2019
%e A135221 First few rows of the triangle:
%e A135221   1;
%e A135221   0, 1;
%e A135221   2, 1,  1;
%e A135221   0, 4,  2,  1;
%e A135221   2, 3,  7,  3,  1;
%e A135221   0, 6,  9, 11,  4,  1;
%e A135221   2, 5, 16, 19, 16,  5,  1;
%e A135221   0, 8, 20, 36, 34, 22,  6, 1;
%e A135221   2, 7, 29, 55, 71, 55, 29, 7, 1;
%e A135221 ...
%p A135221 seq(seq( `if`(k=n, 1, binomial(n,k) + (-1)^(n-k)), k=0..n), n=0..12); # _G. C. Greubel_, Nov 20 2019
%t A135221 T[n_, k_]:= T[n, k]= If[k==n, 1, Binomial[n, k] + (-1)^(n-k)] ;
%t A135221 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)
%o A135221 (PARI) T(n,k) = if(k==n, 1, binomial(n,k) + (-1)^(n-k)); \\ _G. C. Greubel_, Nov 20 2019
%o A135221 (Magma) T:= func< n,k | k eq n select 1 else Binomial(n,k) +(-1)^(n-k) >;
%o A135221 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2019
%o A135221 (Sage)
%o A135221 def T(n, k):
%o A135221     if (k==n): return 1
%o A135221     else: return binomial(n,k) + (-1)^(n-k)
%o A135221 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 20 2019
%o A135221 (GAP)
%o A135221 T:= function(n,k)
%o A135221     if k=n then return 1;
%o A135221     else return Binomial(n,k) + (-1)^(n-k);
%o A135221     fi; end;
%o A135221 Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 20 2019
%Y A135221 Cf. A007318, A051049.
%K A135221 nonn,tabl
%O A135221 0,4
%A A135221 _Gary W. Adamson_, Nov 23 2007
%E A135221 More terms added by _G. C. Greubel_, Nov 20 2019