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%I #15 Sep 08 2022 08:45:30
%S 1,1,1,1,5,1,1,9,9,1,1,13,21,13,1,1,17,37,37,17,1,1,21,57,77,57,21,1,
%T 1,25,81,137,137,81,25,1,1,29,109,221,277,221,109,29,1,1,33,141,333,
%U 501,501,333,141,33,1,1,37,177,477,837,1005,837,477,177,37,1
%N Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.
%C Row sums = A131062: (1, 2, 7, 20, 49, 110, 235, ...); the binomial transform of (1, 1, 4, 4, 4, ...).
%C Triangle equals 4*A007318 - 3*A000012 as infinite lower triangular matrices. - _Emeric Deutsch_, Jun 21 2007
%H G. C. Greubel, <a href="/A131061/b131061.txt">Rows n = 0..100 of triangle, flattened</a>
%F G.f.:(1 - z - t*z + 4*t*z^2)/((1-z)*(1-t*z)*(1-z-t*z)). - _Emeric Deutsch_, Jun 21 2007
%e First few rows of the triangle are
%e 1;
%e 1, 1;
%e 1, 5, 1;
%e 1, 9, 9, 1;
%e 1, 13, 21, 13, 1;
%e 1, 17, 37, 37, 17, 1;
%e 1, 21, 57, 77, 57, 21, 1;
%e ...
%p T := proc (n, k) if k <= n then 4*binomial(n, k)-3 else 0 end if end proc; for n from 0 to 10 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - _Emeric Deutsch_, Jun 21 2007
%t Table[4*Binomial[n, k] -3, {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 12 2020 *)
%o (Magma) [4*Binomial(n, k) -3: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Mar 12 2020
%o (Sage) [[4*binomial(n, k) -3 for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 12 2020
%Y Cf. A109128, A123203, A131060, A131063, A131064, A131065, A131066, A131067, A131068.
%K nonn,tabl
%O 0,5
%A _Gary W. Adamson_, Jun 13 2007
%E More terms from _Emeric Deutsch_, Jun 21 2007
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