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A034852 Rows of (Pascal's triangle - Losanitsch's triangle) (n >= 0, k >= 0). 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

REFERENCES

S. J. Cyvin et al., 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.

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. 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

EXAMPLE

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

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Last modified December 11 07:41 EST 2019. Contains 329914 sequences. (Running on oeis4.)