cp's OEIS Frontend

A377041 First term of the n-th differences of the squarefree numbers. Inverse zero-based binomial transform of A005117.

%I A377041 #5 Oct 19 2024 08:31:05
%S A377041 1,1,0,1,-3,6,-8,3,22,-92,252,-578,1189,-2255,3991,-6617,10245,-14626,
%T A377041 18666,-19635,12104,13090,-69122,171478,-332718,552138,-798629,982514,
%U A377041 -901485,116219,2351842,-8715135,23856206,-57926011,130281064,-273804584,535390333
%N A377041 First term of the n-th differences of the squarefree numbers. Inverse zero-based binomial transform of A005117.
%F A377041 The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:
%F A377041   p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)
%t A377041 q=Select[Range[100],SquareFreeQ];
%t A377041 t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]/2}]
%Y A377041 The version for primes is A007442, noncomposites A030016, composites A377036.
%Y A377041 This is the first column of A377038.
%Y A377041 For nonsquarefree numbers we have A377049.
%Y A377041 For prime-powers we have A377054.
%Y A377041 A000040 lists the primes, differences A001223, seconds A036263.
%Y A377041 A005117 lists the squarefree numbers, complement A013929 (differences A078147).
%Y A377041 A073576 counts integer partitions into squarefree numbers, factorizations A050320.
%Y A377041 A377042 gives first position of 0 in each row of A377038.
%Y A377041 Cf. A053797, A053806, A061398, A072284, A076259, A120992, A376311, A376590, A376591, A377040, A377046.
%K A377041 sign
%O A377041 0,5
%A A377041 _Gus Wiseman_, Oct 18 2024