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%I #6 Dec 14 2024 10:51:20
%S 1,1,0,1,0,0,2,1,1,1,2,0,-1,-2,-3,3,1,1,2,4,7,4,1,0,-1,-3,-7,-14,5,1,
%T 0,0,1,4,11,25,6,1,0,0,0,-1,-5,-16,-41,8,2,1,1,1,1,2,7,23,64,10,2,0,
%U -1,-2,-3,-4,-6,-13,-36,-100,12,2,0,0,1,3,6,10,16,29,65,165
%N Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the strict partition numbers A000009.
%e As a table (read by antidiagonals downward):
%e n=0: n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8:
%e ----------------------------------------------------------
%e k=0: 1 1 1 2 2 3 4 5 6
%e k=1: 0 0 1 0 1 1 1 1 2
%e k=2: 0 1 -1 1 0 0 0 1 0
%e k=3: 1 -2 2 -1 0 0 1 -1 0
%e k=4: -3 4 -3 1 0 1 -2 1 1
%e k=5: 7 -7 4 -1 1 -3 3 0 -3
%e k=6: -14 11 -5 2 -4 6 -3 -3 7
%e k=7: 25 -16 7 -6 10 -9 0 10 -14
%e k=8: -41 23 -13 16 -19 9 10 -24 24
%e k=9: 64 -36 29 -35 28 1 -34 48 -34
%e As a triangle (read by rows):
%e 1
%e 1 0
%e 1 0 0
%e 2 1 1 1
%e 2 0 -1 -2 -3
%e 3 1 1 2 4 7
%e 4 1 0 -1 -3 -7 -14
%e 5 1 0 0 1 4 11 25
%e 6 1 0 0 0 -1 -5 -16 -41
%e 8 2 1 1 1 1 2 7 23 64
%t nn=20;
%t t=Table[Take[Differences[PartitionsQ/@Range[0,2nn],k],nn],{k,0,nn}];
%t Table[t[[j,i-j+1]],{i,nn/2},{j,i}]
%Y Rows are: A000009 (k=0), A087897 (k=1, without first term), A378972 (k=2).
%Y For primes we have A095195 or A376682.
%Y For partitions we have A175804.
%Y First column is A293467 (up to sign).
%Y For composites we have A377033.
%Y For squarefree numbers we have A377038.
%Y For nonsquarefree numbers we have A377046.
%Y For prime powers we have A377051.
%Y Position of first zero in each row is A377285.
%Y Triangle's row-sums are A378970, absolute A378971.
%Y A000009 counts strict integer partitions, differences A087897, A378972.
%Y A000041 counts integer partitions, differences A002865, A053445.
%Y Cf. A047966, A098859, A225486, A325244, A325282.
%Y Cf. A008284, A116608, A325242, A225485 or A325280.
%K sign,tabl,new
%O 0,7
%A _Gus Wiseman_, Dec 13 2024