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A058399
Triangle of partial row sums of partition triangle A008284.
13
1, 2, 1, 3, 2, 1, 5, 4, 2, 1, 7, 6, 4, 2, 1, 11, 10, 7, 4, 2, 1, 15, 14, 11, 7, 4, 2, 1, 22, 21, 17, 12, 7, 4, 2, 1, 30, 29, 25, 18, 12, 7, 4, 2, 1, 42, 41, 36, 28, 19, 12, 7, 4, 2, 1, 56, 55, 50, 40, 29, 19, 12, 7, 4, 2, 1, 77, 76, 70, 58, 43, 30, 19, 12, 7, 4, 2, 1, 101, 100, 94, 80, 62
OFFSET
1,2
COMMENTS
T(n,m) is also the number of m-th largest elements in all partitions of n. - Omar E. Pol, Feb 14 2012
It appears that reversed rows converge to A000070. - Omar E. Pol, Mar 10 2012
The row sums give A006128. - Omar E. Pol, Mar 26 2012
T(n,m) is also the number of regions traversed by the m-th column of the section model of partitions with n sections (Cf. A135010, A206437). - Omar E. Pol, Apr 20 2012
LINKS
FORMULA
T(n, m) = Sum_{k=m..n} A008284(n, k).
G.f. for m-th column: Sum_{n>=1} x^(n)/Product_{k=1..n+m-1} (1 - x^k).
T(n, m) = Sum_{k=1..n} A207379(k, m). - Omar E. Pol, Apr 22 2012
EXAMPLE
From Omar E. Pol, Mar 10 2012: (Start)
Triangle begins:
1;
2, 1;
3, 2, 1;
5, 4, 2, 1;
7, 6, 4, 2, 1;
11, 10, 7, 4, 2, 1;
15, 14, 11, 7, 4, 2, 1;
22, 21, 17, 12, 7, 4, 2, 1;
30, 29, 25, 18, 12, 7, 4, 2, 1;
42, 41, 36, 28, 19, 12, 7, 4, 2, 1;
56, 55, 50, 40, 29, 19, 12, 7, 4, 2, 1;
77, 76, 70, 58, 43, 30, 19, 12, 7, 4, 2, 1;
(End)
MAPLE
b:= proc(n, k) option remember;
`if`(n=0, 1, `if`(k<1, 0, add(b(n-j*k, k-1), j=0..n/k)))
end:
T:= (n, m)-> b(n, n) -b(n, m-1):
seq (seq (T(n, m), m=1..n), n=1..15); # Alois P. Heinz, Apr 20 2012
MATHEMATICA
t[n_, m_] := Sum[ IntegerPartitions[n, {k}] // Length, {k, m, n}]; Table[t[n, m], {n, 1, 13}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 21 2013 *)
CROSSREFS
Columns 1-5: A000041(n), A000065(n+1), A004250(n+2), A035300(n-1), A035301(n-1), n >= 1.
Cf. A008284.
Sequence in context: A119441 A347227 A322083 * A209434 A207611 A320973
KEYWORD
nonn,tabl
AUTHOR
Wolfdieter Lang, Dec 11 2000
STATUS
approved