[seqfan] puzzled by central binomials in A094796
Richard J. Mathar
mathar at mpia-hd.mpg.de
Fri Nov 15 11:40:39 CET 2019
I don't understand A094796. By definition there
is a triangle/array D(n,k) where the column k=0 is A000984,
the column k=1 is A051924, and by my standards the
array D(n,k) starts
1 ;
2 4 ;
6 14 36 ;
20 50 132 358 ;
70 182 490 1346 3748 ;
252 672 1836 5094 14288 40404 ;
924 2508 6930 19382 54692 155388 443886 ;
3432 9438 26312 74074 210080 599274 1717536 4941654 ;
12870 35750 100386 284154 809354 2316810 6659190 19206174 55555236 ;
There seems to be a claim in A094796 that D(n,k) divided
by binomial(2n,n) and product (n+i),i=1..k) gives an integer
array. But my result here is that P(n,k) := D(n,k)/[binomial(2n,n)*product(n+i,i=1..k))]
gives fractions, so column k of P(n,k) cannot be represented by a polynomial
with integer coefficients::
1
1 1
1 7/9 1/2
1 5/8 33/100 179/1200
1 13/25 7/30 673/7350 937/29400
1 4/9 17/98 283/4704 893/47628 481/90720
In Maple:
Dnk := proc(n,k)
option remember;
if k < 0 then
0 ;
elif k = 0 then
binomial(2*n,n) ;
else
procname(n+1,k-1)-procname(n,k-1) ;
end if;
end proc:
A094796 := proc(n,k)
local i;
Dnk(n,k)/binomial(2*n,n)/mul(n+i,i=1..k) ;
end proc:
for n from 0 to 8 do
for k from 0 to n do
# printf("%4d ",Dnk(n,k)) ;
printf("%a ",A094796(n,k)) ;
end do:
printf("\n") ;
end do:
Where is the error?
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