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 A034851 Rows of Losanitsch's triangle T(n, k), n >= 0, 0 <= k <= n. 67
 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, 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, 110, 126, 110, 60, 25, 5, 1, 1, 6, 30, 85, 170, 236, 236, 170, 85, 30, 6, 1, 1, 6, 36, 110, 255 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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 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 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 Alternating row sums are 1,0,1,0,2,0,4,0,8,0,16,0, ... - Gerald McGarvey, Oct 20 2008 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 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 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 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 From Álvar Ibeas, Jun 01 2020: (Start) 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. 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. (End) Named after the Serbian chemist, politician and diplomat Simeon Milivoje "Sima" Lozanić (1847-1935). - Amiram Eldar, Jun 10 2021 LINKS Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened F. Al-Kharousi, R. Kehinde, and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Vol. 58, No. 3 (2014), pp. 363-375. Tewodros Amdeberhan, Mahir Bilen Can and Victor H. Moll, Broken bracelets, Molien series, paraffin wax and the elliptic curve of conductor 48, SIAM Journal of Discrete Math., Vol. 25, No. 4 (2011), p. 1843-1859; arXiv preprint, arXiv:1106.4693 [math.CO], 2011. See Theorem 2.8. Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017. Sahir Gill, Bounds for Region Containing All Zeros of a Complex Polynomial, International Journal of Mathematical Analysis Vol. 12, No. 7 (2018), pp. 325-333. Stephen G. Hartke and A. J. Radcliffe, Signatures of Strings, Annals of Combinatorics, Vol. 17, No. 1 (March, 2013), pp. 131-150. Rethinasamy K. Kittappa, Combinatorial enumeration of rectangular kolam designs of the Tamil land, Abstracts Amer. Math. Soc., Vol. 29, No. 1 (2008), p. 24 (Abstract 1035-05-543). Wolfdieter Lang, Illustration of initial rows of triangle. S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. Vol. 30 (1897), pp. 1917-1926. S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy) Ministry of Foreign Affairs of Serbia, List of the Ministers for Foreign Affairs Since the Forming of the First Government in 1811-Sima Lozanic. N. J. A. Sloane, Classic Sequences. Eric Weisstein's World of Mathematics, Losanitsch's Triangle. Wikipedia, Sima Lozanic. FORMULA T(n, k) = (1/2) * (A007318(n, k) + A051159(n, k)). 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(A034839(n, m)*x^(2*m), m=0..floor(n/2)) (row polynomials of array Pascal even numbered columns). - Wolfdieter Lang, May 08 2001 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 even, k odd (see Sloane link). T(n, k) = T(n-2, k-2) + T(n-2, k) + C(n-2, k-1), n>1. 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 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 From Christopher Hunt Gribble, Feb 25 2014: (Start) It appears that: T(n,k) = C(n,k)/2,                        n even, k odd; T(n,k) = (C(n,k) + C(n/2,k/2))/2,         n even, k even; T(n,k) = (C(n,k) + C((n-1)/2,(k-1)/2))/2, n odd,  k odd; T(n,k) = (C(n,k) + C((n-1)/2,k/2))/2,     n odd,  k even. (End) EXAMPLE Triangle begins   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, 19, 12,  4,  1;   1,  4, 16, 28, 38, 28, 16,  4,  1;   1,  5, 20, 44, 66, 66, 44, 20,  5,  1; MAPLE 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); MATHEMATICA 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 *) PROG (PARI) {T(n, k) = (1/2) *(binomial(n, k) + binomial(n%2, k%2) * binomial(n\2, k\2))}; /* Michael Somos, Oct 20 1999 */ (Haskell) a034851 n k = a034851_row n !! k a034851_row 0 =  a034851_row 1 = [1, 1] a034851_row n = zipWith (-) (zipWith (+) ( ++ losa) (losa ++ ))                             ( ++ a204293_row (n-2) ++ )    where losa = a034851_row (n-1) a034851_tabl = map a034851_row [0..] -- Reinhard Zumkeller, Jan 14 2012 CROSSREFS Cf. A007318, A034852, A051159, A055138, A102541, A228570, A228572. Columns: A008619, A087811, A005993 - A005995, A018210 - A018214, A062136, A141783. 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). Sums of squares of terms in rows equal A211208. Sequence in context: A276696 A220777 A088855 * A172453 A172479 A339788 Adjacent sequences:  A034848 A034849 A034850 * A034852 A034853 A034854 KEYWORD nonn,tabl,easy,nice AUTHOR EXTENSIONS More terms from James A. Sellers, May 04 2000 Name edited by Johannes W. Meijer, Aug 26 2013 STATUS approved

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Last modified October 19 22:05 EDT 2021. Contains 348095 sequences. (Running on oeis4.)