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%I #149 Oct 19 2023 07:36:14
%S 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,6,6,3,1,1,3,9,10,9,3,1,1,4,12,19,
%T 19,12,4,1,1,4,16,28,38,28,16,4,1,1,5,20,44,66,66,44,20,5,1,1,5,25,60,
%U 110,126,110,60,25,5,1,1,6,30,85,170,236,236,170,85,30,6,1,1,6,36,110,255
%N Rows of Losanitsch's triangle T(n, k), n >= 0, 0 <= k <= n.
%C Sometimes erroneously called "Lossnitsch's triangle". But the author's name is Losanitsch (I have seen the original paper in Chem. Ber.). This is a German version of the Serbian name Lozanic. - _N. J. A. Sloane_, Jun 29 2008
%C For n >= 3, a(n-3,k) is the number of series-reduced (or homeomorphically irreducible) trees which become a path P(k+1) on k+1 nodes, k >= 0, when all leaves are omitted (see illustration). Proof by Pólya's enumeration theorem. - _Wolfdieter Lang_, Jun 08 2001
%C The number of ways to put beads of two colors in a line, but take symmetry into consideration, so that 011 and 110 are considered the same. - Yong Kong (ykong(AT)nus.edu.sg), Jan 04 2005
%C Alternating row sums are 1,0,1,0,2,0,4,0,8,0,16,0,... - _Gerald McGarvey_, Oct 20 2008
%C The triangle sums, see A180662 for their definitions, link Losanitsch's triangle A034851 with several sequences, see the crossrefs. We observe that the Ze3 and Ze4 sums link Losanitsch's triangle with A005683, i.e., _R. K. Guy_'s Twopins game. - _Johannes W. Meijer_, Jul 14 2011
%C T(n-(L-1)k, k) is the number of ways to cover an n-length line by exactly k L-length segments excluding symmetric covers. For L=2 it is corresponds to A102541, for L=3 to A228570 and for L=4 to A228572. - _Philipp O. Tsvetkov_, Nov 08 2013
%C Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X 1 rectangle under all symmetry operations of the rectangle. - _Christopher Hunt Gribble_, Feb 16 2014
%C T(n, k) is the number of non-isomorphic outer planar graphs of order n+3, size n+3+k, and maximum degree k+2. - _Christian Barrientos_, Oct 18 2018
%C From _Álvar Ibeas_, Jun 01 2020: (Start)
%C T(n, k) is the sum of even-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 T(n, k) paths and odd for A034852(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 T(n, k) strings and odd for A034852(n, k) cases.
%C (End)
%C Named after the Serbian chemist, politician and diplomat Simeon Milivoje "Sima" Lozanić (1847-1935). - _Amiram Eldar_, Jun 10 2021
%C T(n, k) is the number of caterpillars with a perfect matching, with 2n+2 vertices and diameter 2n-1-k. - _Christian Barrientos_, Sep 12 2023
%H Reinhard Zumkeller, <a href="/A034851/b034851.txt">Rows n = 0..100 of triangle, flattened</a>
%H F. Al-Kharousi, R. Kehinde, and A. Umar, <a href="http://ajc.maths.uq.edu.au/pdf/58/ajc_v58_p365.pdf">Combinatorial results for certain semigroups of partial isometries of a finite chain</a>, The Australasian Journal of Combinatorics, Vol. 58, No. 3 (2014), pp. 363-375.
%H Tewodros Amdeberhan, Mahir Bilen Can and Victor H. Moll, <a href="http://dx.doi.org/10.1137/110819925">Broken bracelets, Molien series, paraffin wax and the elliptic curve of conductor 48</a>, SIAM Journal of Discrete Math., Vol. 25, No. 4 (2011), p. 1843-1859; <a href="http://arxiv.org/abs/1106.4693">arXiv preprint</a>, arXiv:1106.4693 [math.CO], 2011. See Theorem 2.8.
%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 Sahir Gill, <a href="https://doi.org/10.12988/ijma.2018.8537">Bounds for Region Containing All Zeros of a Complex Polynomial</a>, International Journal of Mathematical Analysis Vol. 12, No. 7 (2018), pp. 325-333.
%H Stephen G. Hartke and A. J. Radcliffe, <a href="http://www.combinatorics.net/Annals/Abstract/17_1_131.aspx">Signatures of Strings</a>, Annals of Combinatorics, Vol. 17, No. 1 (March, 2013), pp. 131-150.
%H Rethinasamy K. Kittappa, <a href="http://jointmathematicsmeetings.org/meetings/national/jmm/1035-05-543.pdf">Combinatorial enumeration of rectangular kolam designs of the Tamil land</a>, Abstracts Amer. Math. Soc., Vol. 29, No. 1 (2008), p. 24 (Abstract 1035-05-543).
%H Wolfdieter Lang, <a href="/A034851/a034851.jpg">Illustration of initial rows of triangle</a>.
%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. Vol. 30 (1897), pp. 1917-1926.
%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)
%H Ministry of Foreign Affairs of Serbia, <a href="http://www.mfa.gov.rs/en/diplomatic-tradition/ministers-through-history">List of the Ministers for Foreign Affairs Since the Forming of the First Government in 1811-Sima Lozanic</a>.
%H N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LosanitschsTriangle.html">Losanitsch's Triangle</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Sima_Lozani%C4%87">Sima Lozanic</a>.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F T(n, k) = (1/2) * (A007318(n, k) + A051159(n, k)).
%F G.f. for k-th column (if formatted as lower triangular matrix a(n, k)): x^k*Pe(floor((k+1)/2), x^2)/(((1-x)^(k+1))*(1+x)^(floor((k+1)/2))), where Pe(n, x^2) := Sum_{m=0..floor(n/2)} A034839(n, m)*x^(2*m) (row polynomials of Pascal array even numbered columns). - _Wolfdieter Lang_, May 08 2001
%F a(n, k) = a(n-1, k-1) + a(n-1, k) - C(n/2-1, (k-1)/2), where the last term is present only if n is even and k is odd (see Sloane link).
%F T(n, k) = T(n-2, k-2) + T(n-2, k) + C(n-2, k-1), n > 1.
%F Let P(n, x, y) = Sum_{m=0..n} a(n, m)*x^m*y^(n-m), then for x > 0, y > 0 we have P(n, x, y) = (x+y)*P(n-1, x, y) for n odd and P(n, x, y) = (x+y)*P(n-1, x, y) - x*y*(x^2+y^2)^((n-2)/2) for n even. - _Gerald McGarvey_, Feb 15 2005
%F T(n, k) = T(n-1, k-1) + T(n-1, k) - A204293(n-2, k-1), 0 < k <= n and n > 1. - _Reinhard Zumkeller_, Jan 14 2012
%F From _Christopher Hunt Gribble_, Feb 25 2014: (Start)
%F It appears that:
%F T(n,k) = C(n,k)/2, n even, k odd;
%F T(n,k) = (C(n,k) + C(n/2,k/2))/2, n even, k even;
%F T(n,k) = (C(n,k) + C((n-1)/2,(k-1)/2))/2, n odd, k odd;
%F T(n,k) = (C(n,k) + C((n-1)/2,k/2))/2, n odd, k even.
%F (End)
%e Triangle begins
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 2, 4, 2, 1;
%e 1, 3, 6, 6, 3, 1;
%e 1, 3, 9, 10, 9, 3, 1;
%e 1, 4, 12, 19, 19, 12, 4, 1;
%e 1, 4, 16, 28, 38, 28, 16, 4, 1;
%e 1, 5, 20, 44, 66, 66, 44, 20, 5, 1;
%p A034851 := proc(n,k) option remember; local t; if k = 0 or k = n then return(1) fi; if n mod 2 = 0 and k mod 2 = 1 then t := binomial(n/2-1,(k-1)/2) else t := 0; fi; A034851(n-1,k-1)+A034851(n-1,k)-t; end: seq(seq(A034851(n, k), k=0..n), n=0..11);
%t t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_, k_] := (Binomial[n, k] + Binomial[Quotient[n, 2], Quotient[k, 2]])/2; Flatten[Table[t[n, k], {n, 0, 12}, {k, 0, n}]](* _Jean-François Alcover_, Feb 07 2012, after PARI *)
%o (PARI) {T(n, k) = (1/2) *(binomial(n, k) + binomial(n%2, k%2) * binomial(n\2, k\2))}; /* _Michael Somos_, Oct 20 1999 */
%o (Haskell)
%o a034851 n k = a034851_row n !! k
%o a034851_row 0 = [1]
%o a034851_row 1 = [1,1]
%o a034851_row n = zipWith (-) (zipWith (+) ([0] ++ losa) (losa ++ [0]))
%o ([0] ++ a204293_row (n-2) ++ [0])
%o where losa = a034851_row (n-1)
%o a034851_tabl = map a034851_row [0..]
%o -- _Reinhard Zumkeller_, Jan 14 2012
%Y Cf. A007318, A034852, A051159, A055138, A102541, A228570, A228572.
%Y Columns: A008619, A087811, A005993 - A005995, A018210 - A018214, A062136, A141783.
%Y Triangle sums (see the comments): A005418 (Row), A011782 (Related to Row2), A102526 (Related to Kn11, Kn12, Kn13, Kn21, Kn22, Kn23), A005207 (Kn3, Kn4), A005418 (Fi1, Fi2), A102543 (Ca1, Ca2), A192928 (Gi1, Gi2), A005683 (Ze3, Ze4).
%Y Sums of squares of terms in rows equal A211208.
%K nonn,tabl,easy,nice
%O 0,8
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, May 04 2000
%E Name edited by _Johannes W. Meijer_, Aug 26 2013
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