A204293 Pascal's triangle interspersed with rows of zeros, and the rows of Pascal's triangle are interspersed with zeros.

1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 3, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 0, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 6, 0, 15, 0, 20

Offset: 0

Author

Reinhard Zumkeller, Jan 14 2012

Keywords

• nonn
• tabl

Comments

Auxiliary array for computing Losanitsch's triangle A034851;

T(n, k) + T(n, k + 2) = T(n + 2, k + 2) for k < n - 1.

Links

• Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
• 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: Losanitsch's Triangle
• Eric Weisstein's World of Mathematics, Losanitsch's Triangle
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A077957 (row sums), A126869 (central terms); A108044, A007318.

Programs

Haskell

a204293 n k = a204293_tabl !! n !! k
a204293_row n = a204293_tabl !! n
a204293_tabl = [1] : [0,0] : f [1] [0,0] where
   f xs ys = xs' : f ys xs' where
     xs' = zipWith (+) ([0,0] ++ xs) (xs ++ [0,0])

Mathematica

t[n_?EvenQ, k_?EvenQ] := Binomial[n/2, k/2]; t[, ] = 0; Flatten[Table[t[n, k], {n, 0, 12}, {k, 0, n}]] (* Jean-François Alcover, Feb 07 2012 *)

Formula

T(n, k) = (1 - n mod 2) * (1 - k mod 2) * binomial(floor(n/2),floor(k/2)).

Extensions

Formula for T(n,k) corrected by Peter Bala, Jul 06 2015