A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0).

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 4, 4, 2, 0, 0, 3, 6, 10, 6, 3, 0, 0, 3, 9, 16, 16, 9, 3, 0, 0, 4, 12, 28, 32, 28, 12, 4, 0, 0, 4, 16, 40, 60, 60, 40, 16, 4, 0, 0, 5, 20, 60, 100, 126, 100, 60, 20, 5, 0, 0, 5, 25, 80, 160, 226, 226, 160, 80, 25, 5, 0, 0, 6, 30, 110, 240

Offset

0,12

Comments

Also number of linear unbranched n-4-catafusenes of C_{2v} symmetry.

Number of n-bead black-white reversible strings with k black beads; also binary grids; string is not palindromic. - Yosu Yurramendi, Aug 08 2008

The first seven columns are A004526, A002620, A006584, A032091, A032092, A032093, A032094. Row sums give essentially A032085. - Yosu Yurramendi, Aug 08 2008

From Álvar Ibeas, Jun 01 2020: (Start)

T(n, k) is the sum of odd-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 A034851(n, k) paths and odd for T(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 A034851(n, k) strings and odd for T(n, k) cases.

(End)

Links

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Johann Cigler, Some remarks on Rogers-Szegö polynomials and Losanitsch's triangle, arXiv:1711.03340 [math.CO], 2017.

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll, Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.

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

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

N. J. A. Sloane, Classic Sequences

Formula

Equals (A007318-A051159)/2. - Yosu Yurramendi, Aug 08 2008

T(n, k) = T(n - 1, k - 1) + T(n - 1, k); except when n is even and k odd, in which case T(n, k) = A034851(n, k) = T(n - 1, k - 1) + A034841(n - 1, k) = A034841(n - 1, k - 1) + T(n - 1, k) = C(n, k) / 2. - Álvar Ibeas, Jun 01 2020

Example

Triangle begins:
  0;
  0 0;
  0 1 0;
  0 1 1 0;
  0 2 2 2 0;
  0 2 4 4 2 0;
  ...

Mathematica

nmax = 12; t[n_?EvenQ, k_?EvenQ] := (Binomial[n, k] - Binomial[n/2, k/2])/ 2; t[n_?EvenQ, k_?OddQ] := Binomial[n, k]/2; t[n_?OddQ, k_?EvenQ] := (Binomial[n, k] - Binomial[(n-1)/2, k/2])/2; t[n_?OddQ, k_?OddQ] := (Binomial[n, k] - Binomial[(n-1)/2, (k-1)/2])/2; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Nov 15 2011, after Yosu Yurramendi *)

Prog

(Haskell)
a034852 n k = a034852_tabl !! n !! k
a034852_row n = a034852_tabl !! n
a034852_tabl = zipWith (zipWith (-)) a007318_tabl a034851_tabl
-- Reinhard Zumkeller, Mar 24 2012

Crossrefs

Cf. A007318, A034851, A051159.

Essentially the same as A034877.

Sequence in context: A181674 A181676 A262048 * A212438 A112790 A329922

Adjacent sequences: A034849 A034850 A034851 * A034853 A034854 A034855

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 04 2000

Status

approved