A192928 -id:A192928 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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 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-(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
T(n, k) is the number of non-isomorphic outer planar graphs of order n+3, size n+3+k, and maximum degree k+2. - Christian Barrientos, Oct 18 2018
From Álvar Ibeas, Jun 01 2020: (Start)
T(n, k) is the sum of even-degree coefficients of the Gaussian polynomial [n, k]_q. The area below a NE lattice path between (0,0) and (k, n-k) is even for T(n, k) paths and odd for A034852(n, k) of them.
For a (non-reversible) string of k black and n-k 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 (1847-1935). - Amiram Eldar, Jun 10 2021
T(n, k) is the number of caterpillars with a perfect matching, with 2n+2 vertices and diameter 2n-1-k. - Christian Barrientos, Sep 12 2023

			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;

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
F. Al-Kharousi, R. Kehinde, and A. Umar, Combinatorial results for certain semigroups of partial isometries of a finite chain, The Australasian Journal of Combinatorics, Vol. 58, No. 3 (2014), pp. 363-375.
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. 1843-1859; arXiv preprint, arXiv:1106.4693 [math.CO], 2011. See Theorem 2.8.
Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.
Sahir Gill, Bounds for Region Containing All Zeros of a Complex Polynomial, International Journal of Mathematical Analysis Vol. 12, No. 7 (2018), pp. 325-333.
Stephen G. Hartke and A. J. Radcliffe, Signatures of Strings, Annals of Combinatorics, Vol. 17, No. 1 (March, 2013), pp. 131-150.
Rethinasamy K. Kittappa, Combinatorial enumeration of rectangular kolam designs of the Tamil land, Abstracts Amer. Math. Soc., Vol. 29, No. 1 (2008), p. 24 (Abstract 1035-05-543).
Wolfdieter Lang, Illustration of initial rows of triangle.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. Vol. 30 (1897), pp. 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926. (Annotated scanned copy)
Ministry of Foreign Affairs of Serbia, List of the Ministers for Foreign Affairs Since the Forming of the First Government in 1811-Sima Lozanic.
Jesse Pajwani, Herman Rohrbach, and Anna M. Viergever, Compactly supported A^1-Euler characteristics of symmetric powers of cellular varieties, arXiv:2404.08486 [math.AG], 2024. See p. 15.
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

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.

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

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);

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 *)

{T(n, k) = (1/2) *(binomial(n, k) + binomial(n%2, k%2) * binomial(n\2, k\2))}; /* Michael Somos, Oct 20 1999 */

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_{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(n-1, k-1) + a(n-1, k) - C(n/2-1, (k-1)/2), where the last term is present only if n is even and k is 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
From Christopher Hunt Gribble, Feb 25 2014: (Start)
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.
(End)

More terms from James Sellers, May 04 2000

Name edited by Johannes W. Meijer, Aug 26 2013

A102541 Triangle read by rows, formed from antidiagonals of Losanitsch's triangle. T(n, k) = A034851(n-k, k), n >= 0 and 0 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 4, 1, 1, 3, 6, 2, 1, 4, 9, 6, 1, 1, 4, 12, 10, 3, 1, 5, 16, 19, 9, 1, 1, 5, 20, 28, 19, 3, 1, 6, 25, 44, 38, 12, 1, 1, 6, 30, 60, 66, 28, 4, 1, 7, 36, 85, 110, 66, 16, 1, 1, 7, 42, 110, 170, 126, 44, 4, 1, 8, 49, 146, 255, 236, 110, 20, 1, 1, 8, 56

Offset: 0

Gerald McGarvey, Feb 24 2005

Row sums A102526 are essentially the same as A001224, A060312 and A068928.
Moving the terms in each column of this triangle, see the example, upwards to row 0 gives Losanitsch's triangle A034851 as a square array. - Johannes W. Meijer, Aug 24 2013
The number of ways to cover n-length line by exactly k 2-length segments excluding symmetric covers. - Philipp O. Tsvetkov, Nov 08 2013
Also the number of equivalence classes of ways of placing k 2 X 2 tiles in an n X 2 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014
T(n, k) is the number of irreducible caterpillars with n+3 edges and diameter k+2. - Christian Barrientos, Apr 05 2020

			The first few rows of triangle T(n, k) are:
n/k: 0, 1, 2, 3
0:   1
1:   1
2:   1, 1
3:   1, 1
4:   1, 2, 1
5:   1, 2, 2
6:   1, 3, 4, 1
7:   1, 3, 6, 2

Cf. A034851, A005685, A005688, A068930, A102526, A102543, A192928.

From Johannes W. Meijer, Aug 24 2013: (Start)
T := proc(n,k) option remember: if n <0 then return(0) fi: if k < 0 or k > floor(n/2) then return(0) fi: A034851(n-k, k) end: 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(T(n, k), k=0..floor(n/2)), n=0..16);  # End first program
T := proc(n,k) option remember: if n < 0 then return(0) fi: if k < 0 or k > floor(n/2) then return(0) fi: if n=0 then return(1) fi: if type(n, even) or type(k, even) then procname(n-1, k) + procname(n-2, k-1) else procname(n-1, k) + procname(n-2, k-1) - binomial((n-3)/2-(k-1)/2, (n-3)/2-(k-1)) fi: end: seq(seq(T(n, k), k=0..floor(n/2)), n=0..16); # End second program (End)

t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2;
t[n_, k_] := (Binomial[n, k] + Binomial[Quotient[n, 2], Quotient[k, 2]])/2;
T[n_, k_] := t[n - k, k];
Table[T[n, k], {n, 0, 16}, {k, 0, Quotient[n, 2]}] // Flatten (* Jean-François Alcover, Jul 21 2022 *)

T(n, k) = A034851(n-k, k), n >= 0 and 0 <= k <= floor(n/2).

T(n, k) = T(n-1, k) + T(n-2, k-1) - C((n-3)/2-(k-1)/2, (n-3)/2-(k-1)) except when n or k even then T(n, k) = T(n-1, k) + T(n-2, k-1) with T(0, 0) = 1, T(n, 0) = 0 for n<0 and T(n, k) = 0 for k < 0 and k > floor(n/2). - Johannes W. Meijer, Aug 24 2013

Definition edited, incorrect formula deleted, keyword corrected and extended by Johannes W. Meijer, Aug 24 2013

A228570 Triangle read by rows, formed from antidiagonals of triangle A102541. T(n, k) = A034851(n-2*k, k), n>= 0 and 0 <= k <= floor(n/3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 3, 4, 1, 4, 6, 1, 1, 4, 9, 2, 1, 5, 12, 6, 1, 5, 16, 10, 1, 1, 6, 20, 19, 3, 1, 6, 25, 28, 9, 1, 7, 30, 44, 19, 1, 1, 7, 36, 60, 38, 3, 1, 8, 42, 85, 66, 12, 1, 8, 49, 110, 110, 28, 1

Offset: 0

Johannes W. Meijer, Aug 26 2013

The row sums of this triangle are A102543. The antidiagonal sums are given by A192928 and the backwards antidiagonal sums are given by A228571.
Moving the terms in each column of this triangle, see the example, upwards to row 0 gives Losanitsch’s triangle A034851 as a square array.
Also the number of equivalence classes of ways of placing k 3 X 3 tiles in an n X 3 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Feb 16 2014

			The first few rows of triangle T(n, k) are:
   n/k: 0,  1,  2,  3
   0:   1
   1:   1
   2:   1
   3:   1,  1
   4:   1,  1
   5:   1,  2
   6:   1,  2,  1
   7:   1,  3,  2
   8:   1,  3,  4
   9:   1,  4,  6,  1
  10:   1,  4,  9,  2
  11:   1,  5, 12,  6

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A034851, A102541, A228572, A102543, A192928, A228571, A005691.

T := proc(n,k) option remember: if n <0 then return(0) fi: if k < 0 or k > floor(n/3) then return(0) fi: A034851(n-2*k, k) end: 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(T(n, k), k=0..floor(n/3)), n=0..18);  # End first program
T := proc(n,k) option remember: if n=0 and k=0 or n=1 and k=0 or n=2 and k=0 then return(1) fi: if k <0 or k > floor(n/3) then return(0) fi: if type(n, even) and type(k, odd) then procname(n-1, k) + procname(n-3, k-1) - binomial((n-4)/2-2*(k-1)/2, (k-1)/2) else procname(n-1, k) + procname(n-3, k-1) fi: end: seq(seq(T(n,k), k=0..floor(n/3)), n=0..18); # End second program

T[n_, k_] := (Binomial[n - 2k, k] + Boole[EvenQ[k] || OddQ[n]] Binomial[(n - 2k - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 0, 20}, {k, 0, Quotient[n, 3]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

T(n,k)={(binomial(n-2*k,k) + (k%2==0||n%2==1)*binomial((n-2*k-n%2)/2,k\2))/2}
for(n=1,20,for(k=0,(n\3), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 30 2017

T(n, k) = A034851(n-2*k, k), n >= 0 and 0 <= k <= floor(n/3).

T(n, k) = T(n-1, k) + T(n-3, k-1) - C((n-4)/2 - 2*(k-1)/2, (k-1)/2) where the last term is present only if n even and k odd; T(0, 0) = 1, T(1, 0) = 1, T(2, 0) = 1, T(n, k) = 0 for n < 0 and T(n, k) = 0 for k < 0 and k > floor(n/3).

A228572 Triangle read by rows, formed from antidiagonals of triangle A228570: T(n,k) = A034851(n-3*k, k) for n >= 0 and 0 <= k <= floor(n/4).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 3, 2, 1, 4, 4, 1, 4, 6, 1, 5, 9, 1, 1, 5, 12, 2, 1, 6, 16, 6, 1, 6, 20, 10, 1, 7, 25, 19, 1, 1, 7, 30, 28, 3, 1, 8, 36, 44, 9, 1, 8, 42, 60, 19, 1, 9, 49, 85, 38, 1, 1, 9, 56, 110, 66, 3

Offset: 0

Johannes W. Meijer, Aug 26 2013

The row sums of this triangle are A192928.
Moving the terms in each column of this triangle, see the example, upwards to row 0 gives  Losanitsch’s triangle A034851 as a square array. Observe A102541 and A228570 for the same phenomenom. The number of zeros in the columns for these three triangles are multiples of 2, 3 and 4 respectively.
Also the number of equivalence classes of ways of placing k 4 X 4 tiles in an n X 4 rectangle under all symmetry operations of the rectangle. - Christopher Hunt Gribble, Apr 24 2015

			The first few rows of triangle T(n, k) are:
   n/k: 0, 1, 2, 3
   0:   1
   1:   1
   2:   1
   3:   1
   4:   1, 1
   5:   1, 1
   6:   1, 2
   7:   1, 2
   8:   1, 3, 1
   9:   1, 3, 2
  10:   1, 4, 4
  11:   1, 4, 6
  12:   1, 5, 9, 1

Andrew Howroyd, Table of n, a(n) for n = 0..989

Cf. A034851, A102541, A228570, A192928.

T := proc(n, k) option remember: if n <0 then return(0) fi: if k < 0 or k > floor(n/4) then return(0) fi: A034851(n-3*k, k) end: 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(T(n, k), k=0..floor(n/4)), n=0..21); # End first program
T := proc(n,k) option remember: if n=0 and k=0 or n=1 and k=0 or n=2 and k=0 or n=3 and k=0 then return(1) fi: if k <0 or k > floor(n/4) then return(0) fi: if type(n, odd) and type(k, odd) then procname(n-1,k) + procname(n-4, k-1) - binomial((n+3)/2 - 3*(k+1)/2 - 1,(k+1)/2-1) else procname(n-1, k) + procname(n-4, k-1)  fi: end: seq(seq(T(n, k), k=0..floor(n/4)), n=0..21); # End second program

T[n_, k_] := (Binomial[n - 3k, k] + Boole[EvenQ[k] || EvenQ[n]]* Binomial[(n - 3k - Mod[k, 2] - Mod[n, 2])/2, Quotient[k, 2]])/2; Table[T[n, k], {n, 0, 20}, {k, 0, Quotient[n, 4]}] // Flatten (* Jean-François Alcover, Oct 06 2017, after Andrew Howroyd *)

T(n,k)={(binomial(n-3*k,k) + (k%2==0||n%2==0)*binomial((n-3*k-k%2-n%2)/2,k\2))/2}
for(n=1,20,for(k=0,(n\4), print1(T(n,k), ", "));print) \\ Andrew Howroyd, May 29 2017

T(n, k) = A034851(n-3*k, k) for n >= 0 and 0 <= k <= floor(n/4).

T(n, k) = T(n-1, k) + T(n-4, k-1) - C((n+3)/2 - 3*(k+1)/2-1, (k+1)/2-1), where the last term is present only if n odd and k odd; T(0, 0) = 1, T(1, 0) = 1, T(2, 0) = 1, T(3, 0) = 1, T(n, k) = 0 for n < 0 and T(n, k) = 0 for k < 0 and k > floor(n/4).

cp's OEIS Frontend

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

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A102541 Triangle read by rows, formed from antidiagonals of Losanitsch's triangle. T(n, k) = A034851(n-k, k), n >= 0 and 0 <= k <= floor(n/2).

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A228570 Triangle read by rows, formed from antidiagonals of triangle A102541. T(n, k) = A034851(n-2*k, k), n>= 0 and 0 <= k <= floor(n/3).

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A228572 Triangle read by rows, formed from antidiagonals of triangle A228570: T(n,k) = A034851(n-3*k, k) for n >= 0 and 0 <= k <= floor(n/4).

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A228571 The backwards antidiagonal sums of triangle A228570.

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