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. - njas, 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

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

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.

Sequence in context: A120423 A075402 A088855 this_sequence A122085 A066287 A059260

Adjacent sequences: A034848 A034849 A034850 this_sequence A034852 A034853 A034854

nonn,tabl,easy,nice,new

njas

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