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A145362
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Lower triangular array, called S1hat(-1), related to partition number array A145361.
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4
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1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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1,1
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COMMENTS
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If in the partition array M31hat(-1):=A145361 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-1). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
The first column is [1,1,0,0,0,...]=A008279(1,n-1), n>=1.
a(n,m) gives the number of partitions of n with m parts, with each part not exceeding 2. - Wolfdieter Lang, Aug 03 2023
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LINKS
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FORMULA
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a(n,m) = sum(product(S1(-1;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-1,n,1)= A008279(1,n-1) = [1,1,0,0,0,...], n>=1.
THe triangle starts in row n with ceiling(n/2) - 1 zeros, and is 1 otherwise. - Wolfdieter Lang, Aug 03 2023
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EXAMPLE
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Triangle begins:
[1];
[1,1];
[0,1,1];
[0,1,1,1];
[0,0,1,1,1];
[0,0,1,1,1,1];
...
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CROSSREFS
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Cf. A004526(n+2), n>=1, (row sums).
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KEYWORD
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AUTHOR
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STATUS
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approved
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