[seqfan] A137777 starts with a gap
Peter Luschny
peter.luschny at googlemail.com
Tue Apr 21 15:07:52 CEST 2009
A137777 read as a triangular sequence is
{-2, 4},
{2, -12, 12},
{0,24, -72, 48},
{-8, 0, 240, -480, 240},
{0, -240, 0, 2400, -3600, 1440}, ...
The authors give offset 1 and keyword tabf.
For me it is obvious that the offset is 0,
the appropriate keyword is tabl and the
sequence starts
{2}
{-2, 4},
{2, -12, 12},
{0,24, -72, 48},
{-8, 0, 240, -480, 240},
{0, -240, 0, 2400, -3600, 1440}, ...
Note that then
A137777(n,0) = 2*A129814(n) for n >= 0.
A137777(n,n) = 2*(n+1)! for n >= 0.
The derivatives of the Bernoulli polynomial
function can be generated with Maple
g := (x,t) -> t^2*exp(x*t)/(exp(t)-1):
dg := (x,t) -> diff(g(x,t),t):
p := convert(series(dg(x,t),t,16),polynom):
seq(print(sort(2*i!^2*coeff(p,t,i))),i=0..8);
The Mathematica code given in A137777 could be
replaced by
Clear[p, b, a]; p[t_] = D[t^2*Exp[x*t]/(Exp[t]-1),{t,1}];
a = Table[CoefficientList[2*n!^2*SeriesCoefficient
[Series[p[t],{t,0,30}],n],x],{n,0,10}]; Flatten[a]
Cheers Peter
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