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

Author

Paul Barry, Sep 23 2004

Keywords

• easy
• nonn

Comments

Diagonal sums of A099037.

Links

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

Crossrefs

Cf. A077948.

Programs

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 *)

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

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