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A194708
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Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (8 + m).
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3
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22, 7, 15, 6, 6, 10, 2, 5, 5, 10, 2, 3, 4, 5, 8, 1, 2, 2, 5, 4, 8, 1, 1, 2, 2, 4, 5, 7, 0, 1, 1, 2, 2, 4, 4, 8, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7
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OFFSET
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1,1
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COMMENTS
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Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 8. For further information see A182703 and A135010.
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LINKS
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FORMULA
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T(k,m) = A182703(8+m,k), with T(k,m) = 0 if k > 8+m.
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EXAMPLE
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Triangle begins:
22,
7, 15,
6, 6, 10,
2, 5, 5, 10,
2, 3, 4, 5, 8,
...
For k = 1 and m = 1: T(1,1) = 22 because there are 22 parts of size 1 in the last section of the set of partitions of 9, since 8 + m = 9, so a(1) = 22.
For k = 2 and m = 1: T(2,1) = 7 because there are seven parts of size 2 in the last section of the set of partitions of 9, since 8 + m = 9, so a(2) = 7.
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PROG
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(PARI) P(n)={my(M=matrix(n, n), d=8); M[1, 1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020
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CROSSREFS
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Always the sum of row k = p(8) = A000041(8) = 22.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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