A034850 - OEIS

%I #18 Feb 24 2018 09:23:55

%S 1,1,2,1,3,1,6,1,5,10,1,6,20,6,1,21,35,7,1,28,70,28,1,9,84,126,36,1,

%T 10,120,252,120,10,1,55,330,462,165,11,1,66,495,924,495,66,1,13,286,

%U 1287,1716,715,78,1,14,364,2002,3432,2002,364,14,1,105,1365,5005,6435

%N Triangular array formed by taking every other term of Pascal's triangle.

%H G. C. Greubel, <a href="/A034850/b034850.txt">Table of n, a(n) for the first 100 rows, flattened</a>

%H D. Dumont and J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">Polynômes d'Euler et les fractions continues de Stieltjes-Rogers</a>, preprint 1996.

%H D. Dumont and J. Zeng, <a href="https://doi.org/10.1023/A:1009759202242">Polynômes d'Euler et les fractions continues de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.

%F a(n) = A007318(2n) if both are regarded as integer sequences. - _Michael Somos_, Feb 11 2004

%e Triangle begins:

%e   1;

%e   1;

%e   2;

%e   1,  3;

%e   1,  6,  1;

%e   5, 10,  1;

%e   6, 20,  6;

%e   1, 21, 35,  7;

%t Table[If[k < 0 || k > (Floor[n/4] + Floor[(n + 1)/4]), 0, Binomial[n, 2*k + Mod[Floor[(n + 1)/2], 2]]], {n, 0, 20}, {k, 0, (Floor[n/4] + Floor[(n + 1)/4])}] // Flatten (* _G. C. Greubel_, Feb 23 2018 *)

%o (PARI) {T(n, k) = if( k<0 || k>n\4 + (n+1)\4, 0, binomial(n, 2*k + (n+1)\2%2))}; /* _Michael Somos_, Feb 11 2004 */

%Y Cf. A007318, A034839.

%K nonn,easy,tabf

%O 0,3

%A _N. J. A. Sloane_