%I A034851
%S A034851 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,6,6,3,1,1,3,9,10,9,3,1,1,4,12,19,19,
%T A034851 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,
%U A034851 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 (n >= 0, k >= 0).
%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 (njas(AT)research.att.com), 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 Polya's 
               enumeration theorem. - Wolfdieter Lang (wl(AT)particle.uni-karlsruhe.de), 
               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 ... . [From Gerald 
               McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 20 2008]
%D A034851 R. K. Kittappa, Combinatorial enumeration of rectangular kolam designs 
               of the Tamil land, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), 
               p. 24 (Abstract 1035-05-543).
%D A034851 S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, 
               Chem. Ber. 30 (1897), 1917-1926.
%H A034851 Author?, <a href="http://www.mfa.gov.yu/History/ministri/SLozanic_e.html">
               Sima Lozanic</a>
%H A034851 W. Lang, <a href="a034851.jpg">Illustration of initial rows of triangle</
               a>
%H A034851 N. J. A. Sloane, <a href="classic.html#LOSS">Classic Sequences</a>
%H A034851 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               LossnitschsTriangle.html">Link to a section of The World of Mathematics.</
               a>
%H A034851 Wikipedia, <a href="http://en.wikipedia.org/wiki/Sima_Lozani%C4%87">Sima 
               Lozanic, Serbian chemist</a>
%H A034851 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%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(A034839(n, m)*x^(2*m), m=0..floor(n/2)) (row 
               polynomials of array Pascal even numbered columns). - Wolfdieter 
               Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 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 even, k odd.
%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 (Gerald.McGarvey(AT)comcast.net), Feb 15 2005
%F A034851 Equals=(A051159+A007318)/2. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), 
               Aug 07 2008
%e A034851 1; 1 1; 1 1 1; 1 2 2 1; 1 2 4 2 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;
%o A034851 (PARI) {T(n, k)= (1/2) *(binomial(n, k)+binomial(n%2, k%2)*binomial(n\2, 
               k\2))}
%Y A034851 T(n, k)= (1/2) *(A007318(n, k)+A051159(n, k)). Cf. A007318, A034852, 
               A051159, A055138.
%Y A034851 Row sums give A005418.
%Y A034851 Cf. A007318, A051159.
%Y A034851 Sequence in context: A113137 A075402 A088855 this_sequence A122085 A066287 
               A059260
%Y A034851 Adjacent sequences: A034848 A034849 A034850 this_sequence A034852 A034853 
               A034854
%K A034851 nonn,tabl,easy,nice
%O A034851 0,8
%A A034851 N. J. A. Sloane (njas(AT)research.att.com).
%E A034851 More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 04 2000