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

Simon Plouffe simon.plouffe at gmail.com
Fri Jun 9 21:45:28 CEST 2017



Hello Thomas,

  perhaps there is a link with recurrences :


A000984 : {(-n-2)*a(n+2)+(6+4*n)*a(n+1),a(0)=1,a(1)=2}
A001449 : 
{(15120+41250*n+41875*n^2+18750*n^3+3125*n^4)*a(n+1)+(-4264*n-1680-4016*n^2-1664*n^3-256*n^4)*a(n+2),a(0)=1,a(1)=5}
A005809 : {(60+81*n+27*n^2)*a(n+1)+(-14*n-12-4*n^2)*a(n+2),a(0)=1,a(1)=3}
A005810 : 
{(840+1712*n+1152*n^2+256*n^3)*a(n+1)+(-222*n-120-135*n^2-27*n^3)*a(n+2),a(0)=1,a(1)=4}

all these sequences are P-recurrent.

best regards.

Simon Plouffe


Le 2017-06-09 à 16:23, Thomas Baruchel a écrit :
> 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,
>




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