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A367955
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Number T(n,k) of partitions of [n] whose block maxima sum to k, triangle T(n,k), n>=0, n<=k<=n*(n+1)/2, read by rows.
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11
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 2, 3, 1, 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1, 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1, 1, 1, 2, 5, 10, 23, 47, 39, 49, 81, 84, 129, 74, 78, 70, 87, 33, 23, 29, 5, 6, 1, 1, 1, 2, 5, 10, 23, 47, 103, 81, 129, 172, 261, 304, 431, 299, 325, 376, 317, 424, 196, 183, 144, 165, 52, 34, 41, 6, 7, 1
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OFFSET
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0,7
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COMMENTS
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Rows and also columns reversed converge to A365441.
T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k < n or k > n*(n+1)/2.
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LINKS
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FORMULA
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Sum_{k=n..n*(n+1)/2} k * T(n,k) = A278677(n+1) for n>=1.
Sum_{k=n..n*(n+1)/2} (k-n) * T(n,k) = A200660(n) for n>=1.
T(n,n) = T(n,n*(n+1)/2) = 1.
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EXAMPLE
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T(4,7) = 5: 123|4, 124|3, 13|24, 14|23, 1|2|34.
T(5,9) = 10: 1234|5, 1235|4, 124|35, 125|34, 134|25, 135|24, 14|235, 15|234, 1|23|45, 1|245|3.
T(5,13) = 3: 1|23|4|5, 1|24|3|5, 1|25|3|4.
T(5,14) = 4: 12|3|4|5, 13|2|4|5, 14|2|3|5, 15|2|3|4.
T(5,15) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
1;
. 1;
. . 1, 1;
. . . 1, 1, 2, 1;
. . . . 1, 1, 2, 5, 2, 3, 1;
. . . . . 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1;
. . . . . . 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1;
...
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MAPLE
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b:= proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
end:
T:= (n, k)-> coeff(b(n, 0), x, k):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, t^i, `if`(t=0, 0, t*b(n, i-1, t))+
(t+1)^max(0, 2*i-n-1)*b(n-i, min(n-i, i-1), t+1)))
end:
T:= (n, k)-> b(k, n, 0):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
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CROSSREFS
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Row n has A000124(n-1) terms (for n>=1).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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