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%I #9 Mar 27 2022 18:25:49
%S 1,1,1,2,1,1,1,3,1,1,2,1,4,1,1,1,4,1,5,1,1,2,1,7,1,6,1,1,1,5,1,11,1,7,
%T 1,1,2,1,11,1,16,1,8,1,1,1,6,1,21,1,22,1,9,1,1,2,1,16,1,36,1,29,1,10,
%U 1,1,1,7,1,36,1,57,1,37,1,11,1,1,2,1,22,1,71,1,85,1,46,1,12,1,1
%N Triangle A049310 + A000012 - I, read by rows.
%C Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28, ...).
%H G. C. Greubel, <a href="/A135222/b135222.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n,k) = A049310(n,k) + A000012(n,k) - Identity matrix, as infinite lower triangular matrices.
%F T(n,k) = 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) ), with T(n,n) = 1. - _G. C. Greubel_, Nov 20 2019
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 2, 1, 1;
%e 1, 3, 1, 1;
%e 2, 1, 4, 1, 1;
%e 1, 4, 1, 5, 1, 1;
%e 2, 1, 7, 1, 6, 1, 1;
%e 1, 5, 1, 11, 1, 7, 1, 1;
%e 2, 1, 11, 1, 16, 1, 8, 1, 1;
%e ...
%p T:= proc(n, k) option remember;
%p if k=n then 1
%p else 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) )
%p fi; end:
%p seq(seq(T(n, k), k=0..n), n=0..15); # _G. C. Greubel_, Nov 20 2019
%t T[n_, k_]:= T[n, k]= If[k==n, 1, 1 + Abs[Simplify[((1+(-1)^(n-k))/2)* Binomial[(n+k)/2, (n-k)/2]*Cos[(n-k)*Pi/2]]] ]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)
%o (Sage)
%o @CachedFunction
%o def T(n, k):
%o if (k==n): return 1
%o else: return 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*pi/2) )
%o [[T(n, k) for k in (0..n)] for n in (0..15)] # _G. C. Greubel_, Nov 20 2019
%Y Cf. A049310, A081659.
%K nonn,tabl
%O 0,4
%A _Gary W. Adamson_, Nov 23 2007
%E More terms added and offset changed by _G. C. Greubel_, Nov 20 2019
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