[seqfan] An interesting behaviour of Binomial(kn,n) sequences

[Thomas Baruchel]

Dear fellows,

here is an interesting observation that may have some interest for
building some generating functions for sequences in the database.

Let's call  a_k(n) = binomial(k*n, n)

The generating function of the sequence a_k is:

     g_k(x) = sum(i=0, infty, a_k(i)*x^i)

Whenever k is an integer, g_k is known to be some hypergeometric function
(it is generally documented in the links below).

   For k=2, see A000984
   For k=3, see A005809
   For k=4, see A005810
   For k=5, see A001449
   etc.

I noticed that (1-g_k(x))^2 involves the following convolution:

   C(k,n) = sum(i=2, n-1, (i-1)*binomial(i*k,i)*binomial(k*(n-i), n-i))

equal to:

(k*(-2 + n)*(((-1 + k)^n*(k/(-1 + k))^(k*n))/2 - binomial(k*n, n) - C(k/(-1 + k), (-1 + k)*n)/(-2*k + (-1 + k)*k*n)))/(-1 + k)

I couldn't find it in the eight volumes/PDF from Gould collecting many binomial
convolutions; thus maybe some of you may find some new implications of it.

This will allow to find a relation between sequences of binomial(k*n,n)
and sequences of binomial(k/(k-1)*n,n).

   For k=3, thus k/(k-1) = 3/2, see A244038

Here is a quick example starting from A244038 and leading to a new generating
function for A005809; for convenience reasons, I am using a session in Maxima:

   /* starting from A244038 (each term divided by 4^k) */
   t: taylor(sum( binomial(3/2*k,k)*x^k, k, 0, 31 ), x, 0, 31);
   /* convolution as described above */
   t2: (t-1)^2 /(2*x^2);
   /* by using the classical trick of evaluating the power series in the roots
      of unity, filter the coefficients and take each (k-1)th term; here k=3, which
      allows to do   (g.f.(-x) + g.f.(x))/2   ; in order to have consecutive
      coefficients, we use sqrt(x) rather than x */
   t3: taylor((subst(-sqrt(x),x,t2)+subst(sqrt(x),x,t2))/2, x,0,15);
   /* solve the 2nd degree equation related to the identity above in order to
      reach the generating function of g_k(x)   */
   t4: (-1 + (x*sqrt(-36 + 81*x + 16*t3*x - 108*t3*x^2))/sqrt(-4*x^2 + 27*x^3))/2;
   /* where t4 is the g.f. of A005809 */

Best regards,

-- 
Thomas Baruchel