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A131061 Triangle read by rows: T(n,k) = 4*binomial(n,k) - 3 for 0 <= k <= n.

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