This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A034851 #155 Jul 02 2025 16:01:56
%S A034851 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 A034851 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 A034851 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 A034851 Rows of Losanitsch's triangle T(n, k), n >= 0, 0 <= k <= n.
%C A034851 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 A034851 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 A034851 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 A034851 Alternating row sums are 1,0,1,0,2,0,4,0,8,0,16,0,... - _Gerald McGarvey_, Oct 20 2008
%C A034851 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 A034851 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 A034851 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 A034851 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 A034851 From _Álvar Ibeas_, Jun 01 2020: (Start)
%C A034851 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 A034851 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 A034851 (End)
%C A034851 Named after the Serbian chemist, politician and diplomat Simeon Milivoje "Sima" Lozanić (1847-1935). - _Amiram Eldar_, Jun 10 2021
%C A034851 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 A034851 Reinhard Zumkeller, <a href="/A034851/b034851.txt">Rows n = 0..100 of triangle, flattened</a>
%H A034851 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 A034851 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 A034851 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 A034851 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 A034851 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 A034851 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 A034851 Wolfdieter Lang, <a href="/A034851/a034851.jpg">Illustration of initial rows of triangle</a>.
%H A034851 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 A034851 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 A034851 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 A034851 Jesse Pajwani, Herman Rohrbach, and Anna M. Viergever, <a href="https://arxiv.org/abs/2404.08486">Compactly supported A^1-Euler characteristics of symmetric powers of cellular varieties</a>, arXiv:2404.08486 [math.AG], 2024. See p. 15.
%H A034851 N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>.
%H A034851 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LosanitschsTriangle.html">Losanitsch's Triangle</a>.
%H A034851 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sima_Lozani%C4%87">Sima Lozanic</a>.
%H A034851 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A034851 T(n, k) = (1/2) * (A007318(n, k) + A051159(n, k)).
%F A034851 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 A034851 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 A034851 T(n, k) = T(n-2, k-2) + T(n-2, k) + C(n-2, k-1), n > 1.
%F A034851 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 A034851 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 A034851 From _Christopher Hunt Gribble_, Feb 25 2014: (Start)
%F A034851 It appears that:
%F A034851 T(n,k) = C(n,k)/2, n even, k odd;
%F A034851 T(n,k) = (C(n,k) + C(n/2,k/2))/2, n even, k even;
%F A034851 T(n,k) = (C(n,k) + C((n-1)/2,(k-1)/2))/2, n odd, k odd;
%F A034851 T(n,k) = (C(n,k) + C((n-1)/2,k/2))/2, n odd, k even.
%F A034851 (End)
%e A034851 Triangle begins
%e A034851 1;
%e A034851 1, 1;
%e A034851 1, 1, 1;
%e A034851 1, 2, 2, 1;
%e A034851 1, 2, 4, 2, 1;
%e A034851 1, 3, 6, 6, 3, 1;
%e A034851 1, 3, 9, 10, 9, 3, 1;
%e A034851 1, 4, 12, 19, 19, 12, 4, 1;
%e A034851 1, 4, 16, 28, 38, 28, 16, 4, 1;
%e A034851 1, 5, 20, 44, 66, 66, 44, 20, 5, 1;
%p A034851 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 A034851 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 A034851 (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 A034851 (Haskell)
%o A034851 a034851 n k = a034851_row n !! k
%o A034851 a034851_row 0 = [1]
%o A034851 a034851_row 1 = [1,1]
%o A034851 a034851_row n = zipWith (-) (zipWith (+) ([0] ++ losa) (losa ++ [0]))
%o A034851 ([0] ++ a204293_row (n-2) ++ [0])
%o A034851 where losa = a034851_row (n-1)
%o A034851 a034851_tabl = map a034851_row [0..]
%o A034851 -- _Reinhard Zumkeller_, Jan 14 2012
%Y A034851 Cf. A007318, A034852, A051159, A055138, A102541, A228570, A228572.
%Y A034851 Columns: A008619, A087811, A005993 - A005995, A018210 - A018214, A062136, A141783.
%Y A034851 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 A034851 Sums of squares of terms in rows equal A211208.
%K A034851 nonn,tabl,easy,nice
%O A034851 0,8
%A A034851 _N. J. A. Sloane_
%E A034851 More terms from _James Sellers_, May 04 2000
%E A034851 Name edited by _Johannes W. Meijer_, Aug 26 2013