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
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Crossrefs
Cf. A077948.
Programs
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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
Comments