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A058399 Triangle of partial row sums of partition triangle A008284. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

T(n, m) = sum (A008284(n, k), k=m..n).

G.f. for m-th column: sum(x^(n)/product(1-x^k, k=1..n+m-1), n=1..infty).

T(n, m) = sum {k=1..n} A207379(k, m). - Omar E. Pol, Apr 22 2012

EXAMPLE

Contribution 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-Fran├žois 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: A204927 A119441 A322083 * A209434 A207611 A320973

Adjacent sequences:  A058396 A058397 A058398 * A058400 A058401 A058402

KEYWORD

nonn,tabl

AUTHOR

Wolfdieter Lang, Dec 11 2000

STATUS

approved

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Last modified July 15 00:36 EDT 2020. Contains 335762 sequences. (Running on oeis4.)