[seqfan] Matrix inverses of a recurrence describing Dirichlet convolutions
Mats Granvik
mgranvik at abo.fi
Thu Dec 23 14:09:31 CET 2010
Dear seqfans,
Consider a lower triangular matrix T(n,k) described by T(n,1)=a(n),
k>1: T(n,k) = (sum from i = 1 to k-1 of T(n-i,k-1)) - (sum from i = 1
to k-1 of T(n-i,k)) which is the same as the Dirichlet convolution of
some other sequence b(n).
Call the terms in a(n):
a(1)=a
a(2)=b
a(3)=c
a(4)=d
a(5)=e
a(6)=f
and so on.
Calculate the matrix inverse M(n,k) of the Dirichlet convolution. The
formulas for the first column in M(n,k) can then be expressed by the
terms in a(n).
M(1,1)=a
M(2,1)=-b
M(3,1)=-c+b*b-a*b
M(4,1)=-d+2*c*b-b*b+a*b-a*c-b*b*b+2*a*b*b-a*a*b
M(5,1)=-e+2*d*b-2*c*b+b*b-a*b+c*c-a*d-3*c*b*b+2*b*b*b+4*c*a*b-3*b*a*b+a*a*b-a*a*c+b*b*b*b-3*a*b*b*b+3*a*a*b*b-a*a*a*b
M(6,1)=-f+2*e*b-2*d*b+c*b-b*b+a*b+a*c+2*d*c-c*c-a*e-3*d*b*b+4*d*a*b-5*c*a*b-3*c*c*b+6*b*c*b-2*b*b*b+2*c*a*c+2*a*b*b-a*a*d+4*c*b*b*b-3*b*b*b*b-9*c*a*b*b+7*b*a*b*b+6*c*a*a*b-5*b*a*a*b+a*a*a*b-a*a*a*c-b*b*b*b*b+4*a*b*b*b*b-6*a*a*b*b*b+4*a*a*a*b*b-a*a*a*a*b
Is there any pattern here other than the coefficients of the Pascal
triangle at the end of the formulas?
I calculated this by hand and I was not entirely systematic.
Setting all terms equal to 1 naturally gives the Mobius function while
for example the shortened Fibobacci sequence
a(1)=1,a(2)=2,a(3)=3,a(4)=5,a(5)=8,a(6)=13 gives a variant of the
Möbius function. (1,-2,-1,0,-1,1)
And also numerators of continued fraction convergents to sqrt(2).
a(1)=1,a(2)=3,a(3)=7,a(4)=17,a(5)=41,a(6)=99 give a second variant
of the Möbius function. (1,-3,-1,0,-1,1)
Best regards,
Mats Granvik
http://www.facebook.com/mats.granvik
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