double fun with double Pochhammer

[vdmcc]

Neil,
a simple triangular table, inspired by Bill Dubuque's Binomial theorem for
factorial polynomials.
Please check for goofs!

%I A000001
%S A000001 1,2,1,8,6,1,48,44,12,1,384,400,140,20,1,3840,4384,1800,340,30,1,
%T A000001
46080,56448,25984,5880,700,42,1,645120,836352,420224,108304,15680,
%U A000001
1288,56,1,10321920,14026752,7559936,2153088,359184,36288,2184,72,1
%V A000001
1,-2,1,8,-6,1,-48,44,-12,1,384,-400,140,-20,1,-3840,4384,-1800,340,-30,1,
%W A000001
46080,-56448,25984,-5880,700,-42,1,-645120,836352,-420224,108304,-15680,
%X A000001
1288,-56,1,10321920,-14026752,7559936,-2153088,359184,-36288,2184,-72,1
%N A000001 double Pochhammer triangle, coeff of x in x(x+2)(x+4)..(x+2n-2)
%R A000001
%O A000001 2,2
%K A000001 sign,done
%A A000001 w.meeussen.vdmcc at vandemoortele.be
%D A000001 first column is (2n)!! {1,2,8,48,384...  and the sum of each row
is (2n+1)!! {1,3,15,105,945...
%t A000001 Table[Rest@ CoefficientList[Product[z-k,{k,0,2p-2,2}],z],{p,6}]

in plain :
coeff of p in
p
p(p+2)
p(p+2)(p+4)
p(p+2)(p+4)(p+6)
p(p+2)(p+4)(p+6)(p+8)
...

{{1},
{2,1},
{8,6,1},
{48,44,12,1},
{384,400,140,20,1},
{3840,4384,1800,340,30,1}}
remark that the first column is (2n)!! {1,2,8,48,384...
and the sum of each row is (2n+1)!! {1,3,15,105,945...
you recognise it in Bill's mail "Proof wanted (was: binomial identity)"
                           (p+q)^(/5) ==
(p+q) (2+p+q) (4+p+q) (6+p+q) (8+p+q) ==

p^5*(  1 )+
p^4*( 20 +   5*q) +
p^3*(140 +  80*q +  10*q^2) +
p^2*(400 + 420*q + 120*q^2 + 10*q^3) +
  p*(384 + 800*q + 420*q^2 + 80*q^3 + 5*q^4)+
    (  0 + 384*q + 400*q^2 + 140*q^3 + 20*q^4 + q^5 )

or CoefficientList... ->
{  0, 384, 400, 140, 20, 1}
{384, 800, 420,  80,  5, 0}
{400, 420, 120,  10,  0, 0}
{140,  80,  10,   0,  0, 0}
{ 20,   5,   0,   0,  0, 0}
{  1,   0,   0,   0,  0, 0}



 I am sorry, but the terms
 384,400,140,20
 do not match anything in the table
 I am sorry, but the terms
 20,140,400,384
 do not match anything in the table
wouter.

w.meeussen.vdmcc at vandemoortele.be
tel  +32 (0) 51 33 21 11
fax +32 (0) 51 33 21 75



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