A099038 Diagonal sums of a Krawtchouk triangle.

1, 1, 0, 1, 5, 6, 3, 13, 42, 55, 55, 162, 413, 591, 810, 2001, 4451, 6900, 11091, 24795, 51030, 84337, 147253, 309666, 610695, 1058041, 1928646, 3903175, 7528741, 13480380, 25126093, 49640405, 94739568, 173440389, 326974495, 636424008

Offset

0,5

Comments

Diagonal sums of A099037.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*Sum_{i=0..n} (-1)^i*C(k, i) * C(n-2k, k-i).

Conjecture: n*a(n) -n*a(n-1) +n*a(n-2) +3*(-n+1)*a(n-3) +(-5*n+13)*a(n-4) +(n-3)*a(n-5)=0. - R. J. Mathar, Dec 21 2014

Mathematica

Table[Sum[Binomial[n - k, k]*Sum[(-1)^i*Binomial[k, i]*Binomial[n - 2*k, k - i], {i, 0, n}], {k, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, Dec 31 2017 *)

Prog

(PARI) for(n=0, 30, print1(sum(k=0, floor(n/2), binomial(n-k, k)*sum(i=0, n, (-1)^i*binomial(k, i)*binomial(n-2*k, k-i))), ", ")) \\ G. C. Greubel, Dec 31 2017

Crossrefs

Cf. A077948.

Sequence in context: A328905 A195709 A257837 * A064206 A087197 A221216

Adjacent sequences: A099035 A099036 A099037 * A099039 A099040 A099041

Keyword

easy,nonn

Author

Paul Barry, Sep 23 2004

Status

approved