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A145361
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Characteristic partition array for partitions with parts 1 and 2 only.
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3
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1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to 1 if the partition has parts 1 or 2 only and to 0 otherwise.
First member (K=1) in the family M31hat(-K) of partition number arrays.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
This array is array A144357 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144357/A036040'. E.g. a(4,3)= 1 = 3/3 = A144357(4,3)/A036040(4,3).
If M31hat(-1;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-1):= A145362 .
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LINKS
| W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
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FORMULA
| a(n,k) = product(S1(-1;j,1)^e(n,k,j),j=1..n) with S1(-1;n,1) = A008279(1,n-1) = [1,1,0,0,0,...], n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.
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EXAMPLE
| [1];[1,1];[0,1,1];[0,0,1,1,1];[0,0,0,0,1,1,1];...
a(4,3)= 1 = S1(-1;2,1)^2. The relevant partition of 4 is (2^2).
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CROSSREFS
| A145363 (M31hat(-2)).
Sequence in context: A085357 A132971 A011748 * A130304 A118274 A080909
Adjacent sequences: A145358 A145359 A145360 * A145362 A145363 A145364
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 17 2008
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