This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A377038 #7 Oct 19 2024 08:31:56
%S A377038 1,2,1,3,1,0,5,2,1,1,6,1,-1,-2,-3,7,1,0,1,3,6,10,3,2,2,1,-2,-8,11,1,
%T A377038 -2,-4,-6,-7,-5,3,13,2,1,3,7,13,20,25,22,14,1,-1,-2,-5,-12,-25,-45,
%U A377038 -70,-92,15,1,0,1,3,8,20,45,90,160,252,17,2,1,1,0,-3,-11,-31,-76,-166,-326,-578
%N A377038 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the squarefree numbers.
%C A377038 Row n is the k-th differences of A005117 = the squarefree numbers.
%F A377038 A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A005117(i+k).
%e A377038 Array form:
%e A377038 n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:
%e A377038 ----------------------------------------------------------
%e A377038 k=0: 1 2 3 5 6 7 10 11 13
%e A377038 k=1: 1 1 2 1 1 3 1 2 1
%e A377038 k=2: 0 1 -1 0 2 -2 1 -1 0
%e A377038 k=3: 1 -2 1 2 -4 3 -2 1 1
%e A377038 k=4: -3 3 1 -6 7 -5 3 0 -2
%e A377038 k=5: 6 -2 -7 13 -12 8 -3 -2 3
%e A377038 k=6: -8 -5 20 -25 20 -11 1 5 -5
%e A377038 k=7: 3 25 -45 45 -31 12 4 -10 10
%e A377038 k=8: 22 -70 90 -76 43 -8 -14 20 -19
%e A377038 k=9: -92 160 -166 119 -51 -6 34 -39 28
%e A377038 Triangle form:
%e A377038 1
%e A377038 2 1
%e A377038 3 1 0
%e A377038 5 2 1 1
%e A377038 6 1 -1 -2 -3
%e A377038 7 1 0 1 3 6
%e A377038 10 3 2 2 1 -2 -8
%e A377038 11 1 -2 -4 -6 -7 -5 3
%e A377038 13 2 1 3 7 13 20 25 22
%e A377038 14 1 -1 -2 -5 -12 -25 -45 -70 -92
%e A377038 15 1 0 1 3 8 20 45 90 160 252
%t A377038 nn=9;
%t A377038 t=Table[Take[Differences[NestList[NestWhile[#+1&,#+1,!SquareFreeQ[#]&]&,1,2*nn],k],nn],{k,0,nn}]
%t A377038 Table[t[[j,i-j+1]],{i,nn},{j,i}]
%Y A377038 Row k=0 is A005117.
%Y A377038 Row k=1 is A076259.
%Y A377038 Row k=2 is A376590.
%Y A377038 The version for primes is A095195, noncomposites A376682, composites A377033.
%Y A377038 A version for partitions is A175804, cf. A053445, A281425, A320590.
%Y A377038 Triangle row-sums are A377039, absolute version A377040.
%Y A377038 Column n = 1 is A377041, for primes A007442 or A030016.
%Y A377038 First position of 0 in each row is A377042.
%Y A377038 For nonsquarefree instead of squarefree numbers we have A377046.
%Y A377038 For prime-powers instead of squarefree numbers we have A377051.
%Y A377038 A000040 lists the primes, differences A001223, seconds A036263.
%Y A377038 A005117 lists the squarefree numbers, complement A013929 (differences A078147).
%Y A377038 A073576 counts integer partitions into squarefree numbers, factorizations A050320.
%Y A377038 Cf. A007674, A053797, A053806, A061398, A072284, A075526, A112925, A112926, A120992, A373198, A376311, A376591, A376592.
%K A377038 sign,tabl
%O A377038 0,2
%A A377038 _Gus Wiseman_, Oct 18 2024