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First term of the n-th differences of the squarefree numbers. Inverse zero-based binomial transform of A005117.
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%I #5 Oct 19 2024 08:31:05

%S 1,1,0,1,-3,6,-8,3,22,-92,252,-578,1189,-2255,3991,-6617,10245,-14626,

%T 18666,-19635,12104,13090,-69122,171478,-332718,552138,-798629,982514,

%U -901485,116219,2351842,-8715135,23856206,-57926011,130281064,-273804584,535390333

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

%F The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

%F p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

%t q=Select[Range[100],SquareFreeQ];

%t t=Table[Sum[(-1)^(j-k)*Binomial[j,k]*q[[1+k]],{k,0,j}],{j,0,Length[q]/2}]

%Y The version for primes is A007442, noncomposites A030016, composites A377036.

%Y This is the first column of A377038.

%Y For nonsquarefree numbers we have A377049.

%Y For prime-powers we have A377054.

%Y A000040 lists the primes, differences A001223, seconds A036263.

%Y A005117 lists the squarefree numbers, complement A013929 (differences A078147).

%Y A073576 counts integer partitions into squarefree numbers, factorizations A050320.

%Y A377042 gives first position of 0 in each row of A377038.

%Y Cf. A053797, A053806, A061398, A072284, A076259, A120992, A376311, A376590, A376591, A377040, A377046.

%K sign

%O 0,5

%A _Gus Wiseman_, Oct 18 2024