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%I #40 Mar 15 2024 07:25:51
%S 1,1,1,2,1,1,2,2,1,1,3,2,2,1,1,3,3,2,2,1,1,4,3,3,2,2,1,1,4,4,3,3,2,2,
%T 1,1,5,4,4,3,3,2,2,1,1,5,5,4,4,3,3,2,2,1,1,6,5,5,4,4,3,3,2,2,1,1,6,6,
%U 5,5,4,4,3,3,2,2,1,1,7,6,6,5,5,4,4,3,3,2,2,1,1,7,7,6,6,5,5,4,4,3,3,2,2,1,1
%N Triangle read by rows: row n >= 1 lists first n positive terms of A004526 (integers repeated) in decreasing order.
%C T(n,k) = number of 2-element subsets of {1,2,...,n+2} such that the absolute difference of the elements is k+1, where 1 <= k < = n. E.g., T(7,3) = 3, the subsets are {1,5}, {2,6}, and {3,7}. - _Christian Barrientos_, Jun 27 2022
%H G. C. Greubel, <a href="/A115514/b115514.txt">Rows n = 1..100 of the triangle, flattened</a>
%F Sum_{k=1..n} T(n, k) = A002620(n+1) (row sums). - _Gary W. Adamson_, Oct 25 2007
%F T(n, k) = [x^k] p(n), where p(n) are partial Gaussian polynomials (A008967) defined by p(n) = Sum_{k=0..n} Sum_{j=0..n-k} even(k)*x^j, and even(k) = 1 if k is even and otherwise 0. We assume offset 0. - _Peter Luschny_, Jun 03 2021
%F T(n, k) = floor((n+2-k)/2). - _Christian Barrientos_, Jun 27 2022
%F From _G. C. Greubel_, Mar 14 2024: (Start)
%F T(n, k) = A128623(n, k)/n.
%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A142150(n+1).
%F Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A008805(n-1).
%F Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = A002265(n+3). (End)
%e Triangle begins as, for n >= 1, 1 <= k <= n,
%e 1;
%e 1, 1;
%e 2, 1, 1;
%e 2, 2, 1, 1;
%e 3, 2, 2, 1, 1;
%e 3, 3, 2, 2, 1, 1;
%e 4, 3, 3, 2, 2, 1, 1;
%e ...
%p # Assuming offset 0:
%p Even := n -> (1 + (-1)^n)/2: # Iverson's even.
%p p := n -> add(add(Even(k)*x^j, j = 0..n-k), k = 0..n):
%p for n from 0 to 9 do seq(coeff(p(n), x, k), k=0..n) od; # _Peter Luschny_, Jun 03 2021
%t Table[Floor[(n-k+2)/2], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 14 2024 *)
%o (Magma) [Floor((n-k+2)/2): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 14 2024
%o (SageMath) flatten([[(n-k+2)//2 for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Mar 14 2024
%Y Cf. A000012, A000073, A004526, A008805, A008967.
%Y Cf. A002620 (row sums), A008805 (diagonal sums), A142150 (alternating row sums)
%K nonn,tabl
%O 1,4
%A _Roger L. Bagula_, Mar 07 2006
%E Edited by _N. J. A. Sloane_, Mar 23 2008 and Dec 15 2017
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