[seqfan] Picard-Fuchs Triangle for C(k*n,n)
bradklee at gmail.com
Mon Jul 2 10:05:50 CEST 2018
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,
perhaps we should just enter the triangle starting with:
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''
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.
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