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A034851 Rows of Losanitsch's triangle T(n, k), n >= 0, 0 <= k <= n. 66
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 Polya'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

REFERENCES

F. Al-Kharousi, R. Kehinde, A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Volume 58 (3) (2014), 363-375.

T. Amdeberhan, M. Can and V. Moll, "Broken bracelets, Molien series, paraffin wax and the elliptic curve 48a4," SIAM Journal of Discrete Math., v.25, 2011, p. 1843. See Theorem 2.8.

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

LINKS

Reinhard Zumkeller, Rows n=0..100 of triangle, flattened

T. Amdeberhan, M. B. Can, and V. H. Moll, Broken bracelets, Molien series, paraffin wax and an elliptic curve of conductor 48 see page 6

Stephen G. Hartke and A. J. Radcliffe, Signatures of Strings, Annals of Combinatorics 17 (1) pp.131-150 March, 2013.

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).

W. Lang, Illustration of initial rows of triangle

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

Index entries for sequences related to trees

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

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.

- Christopher Hunt Gribble, Feb 25 2014

EXAMPLE

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

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 = [1]

a034851_row 1 = [1, 1]

a034851_row n = zipWith (-) (zipWith (+) ([0] ++ losa) (losa ++ [0]))

                            ([0] ++ a204293_row (n-2) ++ [0])

   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: A075402 A220777 A088855 * A172453 A172479 A122085

Adjacent sequences:  A034848 A034849 A034850 * A034852 A034853 A034854

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane

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 April 19 05:22 EDT 2014. Contains 240738 sequences.