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

0,13

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

Formula

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

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 *)

Prog

(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])

Crossrefs

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

Sequence in context: A037047 A118917 A325045 * A206479 A219484 A060396

Adjacent sequences: A204290 A204291 A204292 * A204294 A204295 A204296

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Jan 14 2012

Extensions

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

Status

approved