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A034851
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Rows of Losanitsch's triangle (n >= 0, k >= 0).
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29
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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
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0,8
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COMMENTS
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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 ... . [From 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]
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REFERENCES
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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.
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).
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
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LINKS
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_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
Author?, Sima Lozanic
W. Lang, Illustration of initial rows of triangle
N. J. A. Sloane, Classic Sequences
Eric Weisstein's World of Mathematics, Losanitsch's Triangle
Wikipedia, Sima Lozanic, Serbian chemist
Index entries for sequences related to trees
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FORMULA
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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.
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
Equals=(A051159+A007318)/2. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008
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]
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EXAMPLE
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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
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MAPLE
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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;
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MATHEMATICA
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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}]](* From Jean-François Alcover, Feb 07 2012, after Pari *)
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PROG
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(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
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CROSSREFS
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T(n, k)= (1/2) *(A007318(n, k)+A051159(n, k)). Cf. A007318, A034852, A051159, A055138.
Row sums give A005418.
Cf. A007318, A051159.
Columns: A008619, A087811, A005993 - A005995, A018210 - A018214, A062136, A141783. [Gary W. Adamson, Dec 15 2010]
Triangle sums (see the comments): A005418 (Row1), 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). [Johannes W. Meijer, Jul 14 2011]
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
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KEYWORD
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nonn,tabl,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from James A. Sellers, May 04 2000
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STATUS
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approved
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