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A181434 First column in matrix inverse of a mixed convolution of A052542.

%I A181434 #35 Oct 06 2017 07:43:26
%S A181434 1,-3,-1,0,-1,1,-1,0,0,1,-1,0,-1,1,1,0,-1,0,-1,0,1,1,-1,0,0,1,0,0,-1,
%T A181434 -1,-1,0,1,1,1,0,-1,1,1,0,-1,-1,-1,0,0,1,-1,0,0,0,1,0,-1,0,1,0,1,1,-1,
%U A181434 0,-1,1,0,0,1,-1,-1,0,1,-1,-1,0,-1,1,0,0,1
%N A181434 First column in matrix inverse of a mixed convolution of A052542.
%C A181434 It appears that except for the second term, the sequence is identical to the Möbius function.
%C A181434 Explicit numeric calculation confirms this up to at least n=1085. - _R. J. Mathar_, Oct 06 2017
%H A181434 R. J. Mathar, <a href="/A181434/b181434.txt">Table of n, a(n) for n = 1..1085</a>
%F A181434 From _Mats Granvik_, Sep 16 2017: (Start)
%F A181434 a(n) as the matrix inverse of a mixed convolution: Let c = 2 and let the sequence b be defined by the recurrence: b(1) = 1, b(2) = c, b(3) = c^2; for n >= 4, b(n) = c*b(n-1) + b(n-2), so b(n) = A052542(n-1), and let the lower triangular matrix A be: If n >= k then A(n,k) = b(n - k + 1) else A(n,k) = 0, and let B be the lower triangular matrix A051731. Then the matrix inverse (A.B)^-1 will have a(n) as its first column.
%F A181434 The matrix product T = A.B can be defined as follows: Let c = 2 and the sequence b be defined by the recurrence b(0) = 1, b(1) = 1; for b >= 2, b(n) = c*b(n - 1) + b(n - 2), so b(n) = A001333(n); and let T be the lower triangular matrix defined by the recurrence: T(n, 1) = If n >= 1 then T(n, 1) = b(n) else T(n, 1) = 0; for k >= 2, T(n, k) = If n >= k then (Sum_{i=1..k-1} T(n - i, k - 1) - T(n - i, k)) else 0. (Then the matrix inverse of T will have a(n) as its first column.)
%F A181434 (End)
%p A181434 b := proc(n)
%p A181434     option remember;
%p A181434     local c;
%p A181434     c := 2;
%p A181434     if n <= 2 then
%p A181434         n;
%p A181434     elif n = 3 then
%p A181434         c^2 ;
%p A181434     else
%p A181434         c*procname(n-1)+procname(n-2) ;
%p A181434     end if;
%p A181434 end proc:
%p A181434 A := proc(n,k)
%p A181434     if n >= k then
%p A181434         b(n-k+1) ;
%p A181434     else
%p A181434         0 ;
%p A181434     end if;
%p A181434 end proc:
%p A181434 B := proc(n,k)
%p A181434     if modp(n,k) = 0 then
%p A181434         1;
%p A181434     else
%p A181434         0;
%p A181434     end if;
%p A181434 end proc:
%p A181434 AB := proc(n,k)
%p A181434     option remember;
%p A181434     add( A(n,j)*B(j,k),j=1..n) ;
%p A181434 end proc:
%p A181434 ABinv := proc(n,k)
%p A181434     option remember;
%p A181434     if k > n then
%p A181434         0;
%p A181434     elif k = n then
%p A181434         1;
%p A181434     else
%p A181434         -add( AB(n,j)*procname(j,k),j=k..n-1) ;
%p A181434     end if;
%p A181434 end proc:
%p A181434 A181434 := proc(n)
%p A181434     ABinv(n,1) ;
%p A181434 end proc:
%p A181434 for n from 1 do
%p A181434     printf("%d %d\n",n,ABinv(n,1)) ;
%p A181434 end do: # _R. J. Mathar_, Oct 06 2017
%t A181434 Clear[t, n, k, nn, b, A, c]; nn = 77; c = 2; b[0] = 1; b[1] = 1; b[n_] := b[n] = c*b[n - 1] + b[n - 2]; t[n_, 1] = If[n >= 1, b[n], 0]; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}] - Sum[t[n - i, k], {i, 1, k - 1}], 0]; MatrixForm[A = Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]; Inverse[A][[All, 1]] (* _Mats Granvik_, Sep 15 2017 *)
%o A181434 (PARI) A181434(n)=if(n==2,-3,moebius(n)) \\ _M. F. Hasler_, Sep 15 2017. - This program seems to be based on a formula that is so far only conjectural? - _Antti Karttunen_, Oct 06 2017
%Y A181434 Cf. A000129, A001333, A008683, A013946, A178536, A181435.
%K A181434 sign
%O A181434 1,2
%A A181434 _Mats Granvik_, Oct 20 2010