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A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0). 3

%I

%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

%D S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

%D S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.

%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. 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

%e 0; 0 0; 0 1 0; 0 1 1 0; 0 2 2 2 0; 0 2 4 4 2 0; ...

%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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Last modified May 23 06:49 EDT 2018. Contains 304455 sequences. (Running on oeis4.)