# [seqfan] Picard-Fuchs Triangle for C(k*n,n)

Mon Jul 2 10:05:50 CEST 2018

```Hi Seqfans,

On the latest draft for A16991 ( https://oeis.org/draft/A169961 ),
I add that the G.f. of Binomial(12*n,n) satisfies a Picard-Fuchs type
differential equation. The coefficient matrix of the D.E. has two
stripes along the diagonal, assuring a hypergeometric G.f.
Looking at data from Binomial(k*n,n) up to k=12, the two-stripe
phenomena appears fully general, and probably coheres with
patterns in generating functions entered by Ilya Gutkovskiy.

Rather than entering differential equations to all of the following:
A000984, A005809, A005810, A001449, A004355, A004368,
A004381, A169958,A169959,A169960,
perhaps we should just enter the triangle starting with:

1 \\
2  4 \\
6  54  27 \\
24  816  1152  256 \\
120  15000  45000   25000  3125 \\
720  331920  1895400  1976400  583200, 46656 \\
5040  8643600  89029080  155296680  81177810  14823774  823543 \\

Amazing that this triangle isn't already in the OEIS ! ! !

If you are confused, it derives from a sequence of differential equations,
which define the generating functions of C(k*n,n),
C(2*n,n): 2*x*G+(4*x^2 -x)*G' = 0
C(3*n,n): 6*x*G+(54*x^2 -2*x)*G'+(27*x^3-4*x^2)*G''
C(4*n,n):
24*x*G+(816*x^2-6*x)*G'+(1152*x^3-54*x^2)*G''+(256*x^4-27*x^3)*G'''
Etc . . .

We can give a rigorous definition for all k:
Triangle of coefficients -[x^j*G^(j)],  j=1..k-1 in the P.F.D.E. for G.F.
of C(k*n,n), k>1.
( normalized to have integer coefficients with GCD=1 )

Too bad I'm at my limit of 3/3 submissions, ha ha ha.

Cheers,

```