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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. 3

%I #14 Sep 08 2022 08:45:31

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

%C Row sums are in A131269. Reversal = triangle A131270.

%H G. C. Greubel, <a href="/A131268/b131268.txt">Rows n = 0..100 of triangle, flattened</a>

%F 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 Triangle begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 1, 3, 1;

%e 1, 1, 5, 3, 1;

%e 1, 1, 7, 5, 5, 1;

%e 1, 1, 9, 7, 11, 5, 1;

%e 1, 1, 11, 9, 19, 11, 7, 1;

%e 1, 1, 13, 11, 29, 19, 19, 7, 1;

%e 1, 1, 15, 13, 41, 29, 39, 19, 9, 1;

%e 1, 1, 17, 15, 55, 41, 69, 39, 29, 9, 1;

%e 1, 1, 19, 17, 71, 55, 111, 69, 69, 29, 11, 1;

%e 1, 1, 21, 19, 89, 71, 167, 111, 139, 69, 41, 11, 1;

%e ...

%p 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 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 (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 (PARI) T(n,k) = 2*binomial(n- (k+1)\2, k\2) -1; \\ _G. C. Greubel_, Jul 10 2019

%o (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 Cf. A065941, A000012, A131269, A131270.

%K nonn,tabl

%O 0,9

%A _Gary W. Adamson_, Jun 23 2007

%E More terms from _Emeric Deutsch_, Jul 15 2007

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Last modified March 29 06:44 EDT 2024. Contains 371265 sequences. (Running on oeis4.)