0,8

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

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

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 ... . [From Gerald McGarvey (gerald.mcgarvey(AT)comcast.net), Oct 20 2008]

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).

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

Author?, Sima Lozanic

W. Lang, Illustration of initial rows of triangle

N. J. A. Sloane, Classic Sequences

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Wikipedia, Sima Lozanic, Serbian chemist

Index entries for sequences related to trees

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

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.

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 (Gerald.McGarvey(AT)comcast.net), Feb 15 2005

Equals=(A051159+A007318)/2. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008

1; 1 1; 1 1 1; 1 2 2 1; 1 2 4 2 1; ...

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;

(PARI) {T(n, k)= (1/2) *(binomial(n, k)+binomial(n%2, k%2)*binomial(n\2, k\2))}

T(n, k)= (1/2) *(A007318(n, k)+A051159(n, k)). Cf. A007318, A034852, A051159, A055138.

Row sums give A005418.

Cf. A007318, A051159.

Sequence in context: A113137 A075402 A088855 this_sequence A122085 A066287 A059260

Adjacent sequences: A034848 A034849 A034850 this_sequence A034852 A034853 A034854

nonn,tabl,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), May 04 2000