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A131268 Triangle read by rows: T(n,k) = 2*binomial(n-floor((k+1)/2),floor(k/2)) - 1, 0<=k<=n.

%I A131268 #14 Sep 08 2022 08:45:31
%S A131268 1,1,1,1,1,1,1,1,3,1,1,1,5,3,1,1,1,7,5,5,1,1,1,9,7,11,5,1,1,1,11,9,19,
%T A131268 11,7,1,1,1,13,11,29,19,19,7,1,1,1,15,13,41,29,39,19,9,1,1,1,17,15,55,
%U A131268 41,69,39,29,9,1,1,1,19,17,71,55,111,69,69,29,11,1,1,1,21,19,89,71,167,111,139,69,41,11,1
%N A131268 Triangle read by rows: T(n,k) = 2*binomial(n-floor((k+1)/2),floor(k/2)) - 1, 0<=k<=n.
%C A131268 Row sums are in A131269. Reversal = triangle A131270.
%H A131268 G. C. Greubel, <a href="/A131268/b131268.txt">Rows n = 0..100 of triangle, flattened</a>
%F A131268 Equals 2*A065941 - A000012, where A065941 = Pascal's triangle with repeated columns; and A000012 = (1; 1,1; 1,1,1;...) as an infinite lower triangular matrix.
%e A131268 Triangle begins:
%e A131268 1;
%e A131268 1, 1;
%e A131268 1, 1,  1;
%e A131268 1, 1,  3,  1;
%e A131268 1, 1,  5,  3,  1;
%e A131268 1, 1,  7,  5,  5,  1;
%e A131268 1, 1,  9,  7, 11,  5,   1;
%e A131268 1, 1, 11,  9, 19, 11,   7,   1;
%e A131268 1, 1, 13, 11, 29, 19,  19,   7,   1;
%e A131268 1, 1, 15, 13, 41, 29,  39,  19,   9,  1;
%e A131268 1, 1, 17, 15, 55, 41,  69,  39,  29,  9,  1;
%e A131268 1, 1, 19, 17, 71, 55, 111,  69,  69, 29, 11,  1;
%e A131268 1, 1, 21, 19, 89, 71, 167, 111, 139, 69, 41, 11, 1;
%e A131268 ...
%p A131268 T := proc (n, k) options operator, arrow; 2*binomial(n-floor((1/2)*k+1/2), floor((1/2)*k))-1 end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form. - _Emeric Deutsch_, Jul 15 2007
%t A131268 Table[2*Binomial[n -Floor[(k+1)/2], Floor[k/2]] -1, {n,0,14}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 10 2019 *)
%o A131268 (Magma) [2*Binomial(n-Floor((k+1)/2), Floor(k/2))-1: k in [0..n], n in [0..14]]; // _Bruno Berselli_, May 03 2012
%o A131268 (PARI) T(n,k) = 2*binomial(n- (k+1)\2, k\2) -1; \\ _G. C. Greubel_, Jul 10 2019
%o A131268 (Sage) [[2*binomial(n -floor((k+1)/2), floor(k/2)) -1 for k in (0..n)] for n in (0..14)] # _G. C. Greubel_, Jul 10 2019
%Y A131268 Cf. A065941, A000012, A131269, A131270.
%K A131268 nonn,tabl
%O A131268 0,9
%A A131268 _Gary W. Adamson_, Jun 23 2007
%E A131268 More terms from _Emeric Deutsch_, Jul 15 2007