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A176123 Irregular triangle, read by rows, T(n, k) = binomial(n-(k-1),k-1), 1 <= k <= floor(n/2-1).

%I A176123 #12 Sep 08 2022 08:45:52
%S A176123 1,1,1,5,1,6,1,7,15,1,8,21,1,9,28,35,1,10,36,56,1,11,45,84,70,1,12,55,
%T A176123 120,126,1,13,66,165,210,126,1,14,78,220,330,252,1,15,91,286,495,462,
%U A176123 210,1,16,105,364,715,792,462,1,17,120,455,1001,1287,924,330,1,18,136,560,1365,2002,1716,792
%N A176123 Irregular triangle, read by rows, T(n, k) = binomial(n-(k-1),k-1), 1 <= k <= floor(n/2-1).
%C A176123 The row sum is A099572 which has limiting ratio of (1+sqrt(5))/2.
%C A176123 This is Pascal's triangle (A007318) read along upward sloping diagonals and truncated.
%H A176123 G. C. Greubel, <a href="/A176123/b176123.txt">Rows n = 4..100 of triangle, flattened</a>
%e A176123 Triangle begins as:
%e A176123   1;
%e A176123   1;
%e A176123   1,  5;
%e A176123   1,  6;
%e A176123   1,  7, 15;
%e A176123   1,  8, 21;
%e A176123   1,  9, 28,  35;
%e A176123   1, 10, 36,  56;
%e A176123   1, 11, 45,  84,  70;
%e A176123   1, 12, 55, 120, 126;
%e A176123   1, 13, 66, 165, 210, 126;
%p A176123 seq(seq( binomial(n-k+1, k-1), k=1..floor((n-2)/2)), n=4..20); # _G. C. Greubel_, Nov 27 2019
%t A176123 Table[Binomial[n-k+1, k-1], {n,4,20}, {k, Floor[(n-2)/2]}]//Flatten
%o A176123 (PARI) T(n,k) = binomial(n-k+1,k-1);
%o A176123 for(n=4,20, for(k=1, (n-2)\2, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Nov 27 2019
%o A176123 (Magma) [Binomial(n-k+1,k-1): k in [1..Floor((n-2)/2)], n in [4..20]]; // _G. C. Greubel_, Nov 27 2019
%o A176123 (Sage) [[binomial(n-k+1,k-1) for k in (1..floor((n-2)/2))] for n in (4..20)] # _G. C. Greubel_, Nov 27 2019
%o A176123 (GAP) Flat(List([4..20], n-> List([1..Int((n-2)/2)], k-> Binomial(n-k+1,k-1) ))); # _G. C. Greubel_, Nov 27 2019
%Y A176123 Cf. A007318, A099572.
%K A176123 nonn,tabf
%O A176123 4,4
%A A176123 _Roger L. Bagula_, Dec 07 2010
%E A176123 Edited by _N. J. A. Sloane_, Dec 09 2010
%E A176123 More terms added by _G. C. Greubel_, Nov 27 2019