

A034851


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


68



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
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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(n3,k) is the number of seriesreduced (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
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(L1)k, k) is the number of ways to cover an nlength line by exactly k Llength 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
T(n, k) is the number of nonisomorphic outer planar graphs of order n+3, size n+3+k, and maximum degree k+2.  Christian Barrientos, Oct 18 2018
T(n, k) is the sum of evendegree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, nk) is even for T(n, k) paths and odd for A034852(n, k) of them.
For a (nonreversible) string of k black and nk 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.
(End)
Named after the Serbian chemist, politician and diplomat Simeon Milivoje "Sima" Lozanić (18471935).  Amiram Eldar, Jun 10 2021
T(n, k) is the number of caterpillars with a perfect matching, with 2n+2 vertices and diameter 2n1k.  Christian Barrientos, Sep 12 2023


LINKS

Tewodros Amdeberhan, Mahir Bilen Can and Victor H. Moll, Broken bracelets, Molien series, paraffin wax and the elliptic curve of conductor 48, SIAM Journal of Discrete Math., Vol. 25, No. 4 (2011), p. 18431859; arXiv preprint, arXiv:1106.4693 [math.CO], 2011. See Theorem 2.8.
Stephen G. Hartke and A. J. Radcliffe, Signatures of Strings, Annals of Combinatorics, Vol. 17, No. 1 (March, 2013), pp. 131150.


FORMULA

G.f. for kth column (if formatted as lower triangular matrix a(n, k)): x^k*Pe(floor((k+1)/2), x^2)/(((1x)^(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
a(n, k) = a(n1, k1) + a(n1, k)  C(n/21, (k1)/2), where the last term is present only if n is even and k is odd (see Sloane link).
T(n, k) = T(n2, k2) + T(n2, k) + C(n2, k1), n > 1.
Let P(n, x, y) = Sum_{m=0..n} a(n, m)*x^m*y^(nm), then for x > 0, y > 0 we have P(n, x, y) = (x+y)*P(n1, x, y) for n odd and P(n, x, y) = (x+y)*P(n1, x, y)  x*y*(x^2+y^2)^((n2)/2) for n even.  Gerald McGarvey, Feb 15 2005
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((n1)/2,(k1)/2))/2, n odd, k odd;
T(n,k) = (C(n,k) + C((n1)/2,k/2))/2, n odd, k even.
(End)


EXAMPLE

Triangle begins
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;


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/21, (k1)/2) else t := 0; fi; A034851(n1, k1)+A034851(n1, 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}]](* JeanFrançois 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 (n2) ++ [0])
where losa = a034851_row (n1)
a034851_tabl = map a034851_row [0..]


CROSSREFS

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.


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STATUS

approved



