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%I #43 Jul 07 2020 16:24:35
%S 0,0,0,0,1,0,0,1,1,0,0,2,2,2,0,0,2,4,4,2,0,0,3,6,10,6,3,0,0,3,9,16,16,
%T 9,3,0,0,4,12,28,32,28,12,4,0,0,4,16,40,60,60,40,16,4,0,0,5,20,60,100,
%U 126,100,60,20,5,0,0,5,25,80,160,226,226,160,80,25,5,0,0,6,30,110,240
%N Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).
%C Also number of linear unbranched n-4-catafusenes of C_{2v} symmetry.
%C Number of n-bead black-white reversible strings with k black beads; also binary grids; string is not palindromic. - _Yosu Yurramendi_, Aug 08 2008
%C The first seven columns are A004526, A002620, A006584, A032091, A032092, A032093, A032094. Row sums give essentially A032085. - _Yosu Yurramendi_, Aug 08 2008
%C From _Álvar Ibeas_, Jun 01 2020: (Start)
%C T(n, k) is the sum of odd-degree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, n-k) is even for A034851(n, k) paths and odd for T(n, k) of them.
%C For a (non-reversible) string of k black and n-k white beads, consider the minimum number of bead transpositions needed to place the black ones to the left and the white ones to the right (in other words, the number of inversions of the permutation obtained by labeling the black beads by integers 1,...,k and the white ones by k+1,...,n, in the same order they take on the string). It is even for A034851(n, k) strings and odd for T(n, k) cases.
%C (End)
%H Reinhard Zumkeller, <a href="/A034852/b034852.txt">Rows n=0..150 of triangle, flattened</a>
%H Johann Cigler, <a href="https://arxiv.org/abs/1711.03340">Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle</a>, arXiv:1711.03340 [math.CO], 2017.
%H S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, <a href="https://hrcak.srce.hr/177109">Unbranched catacondensed polygonal systems containing hexagons and tetragons</a>, Croatica Chem. Acta, 69 (1996), 757-774.
%H S. M. Losanitsch, <a href="https://doi.org/10.1002/cber.189703002144">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926.
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
%H N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>
%F Equals (A007318-A051159)/2. - _Yosu Yurramendi_, Aug 08 2008
%F T(n, k) = T(n - 1, k - 1) + T(n - 1, k); except when n is even and k odd, in which case T(n, k) = A034851(n, k) = T(n - 1, k - 1) + A034841(n - 1, k) = A034841(n - 1, k - 1) + T(n - 1, k) = C(n, k) / 2. - _Álvar Ibeas_, Jun 01 2020
%e Triangle begins:
%e 0;
%e 0 0;
%e 0 1 0;
%e 0 1 1 0;
%e 0 2 2 2 0;
%e 0 2 4 4 2 0;
%e ...
%t nmax = 12; t[n_?EvenQ, k_?EvenQ] := (Binomial[n, k] - Binomial[n/2, k/2])/ 2; t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_?OddQ, k_?EvenQ] := (Binomial[n, k] - Binomial[(n-1)/2, k/2])/2; t[n_?OddQ, k_?OddQ] := (Binomial[n, k] - Binomial[(n-1)/2, (k-1)/2])/2; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* _Jean-François Alcover_, Nov 15 2011, after Yosu Yurramendi *)
%o (Haskell)
%o a034852 n k = a034852_tabl !! n !! k
%o a034852_row n = a034852_tabl !! n
%o a034852_tabl = zipWith (zipWith (-)) a007318_tabl a034851_tabl
%o -- _Reinhard Zumkeller_, Mar 24 2012
%Y Cf. A007318, A034851, A051159.
%Y Essentially the same as A034877.
%K nonn,tabl,easy,nice
%O 0,12
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 04 2000
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