A212001 Triangle read by rows: T(n,k) = sum of all parts of the last n-k+1 shells of n.

1, 4, 3, 9, 8, 5, 20, 19, 16, 11, 35, 34, 31, 26, 15, 66, 65, 62, 57, 46, 31, 105, 104, 101, 96, 85, 70, 39, 176, 175, 172, 167, 156, 141, 110, 71, 270, 269, 266, 261, 250, 235, 204, 165, 94, 420, 419, 416, 411, 400, 385, 354, 315, 244, 150, 616, 615

Offset

1,2

Comments

The set of partitions of n contains n shells (see A135010). It appears that the last k shells of n contain p(n-k) parts of size k, where p(n) = A000041(n). See also A182703.

Links

Table of n, a(n) for n=1..57.

Formula

T(n,k) = A066186(n) - A066186(k-1).

T(n,k) = Sum_{j=k..n} A138879(j).

Example

For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below the sum of all parts of each set:
--------------------------------------------------------
.  S{1-5}     S{2-5}     S{3-5}     S{4-5}     S{5}
--------------------------------------------------------
.  The        Last       Last       Last       The
.  five       four       three      two        last
.  shells     shells     shells     shells     shell
.  of 5       of 5       of 5       of 5       of 5
--------------------------------------------------------
.
.  5          5          5          5          5
.  3+2        3+2        3+2        3+2        3+2
.  4+1        4+1        4+1        4+1          1
.  2+2+1      2+2+1      2+2+1      2+2+1          1
.  3+1+1      3+1+1      3+1+1        1+1          1
.  2+1+1+1    2+1+1+1      1+1+1        1+1          1
.  1+1+1+1+1    1+1+1+1      1+1+1        1+1          1
. ---------- ---------- ---------- ---------- ----------
.     35         34         31         26         15
.
So row 5 lists 35, 34, 31, 26, 15.
.
Triangle begins:
1;
4,     3;
9,     8,   5;
20,   19,  16,  11;
35,   34,  31,  26,  15;
66,   65,  62,  57,  46,  31;
105, 104, 101,  96,  85,  70,  39;
176, 175, 172, 167, 156, 141, 110,  71;
270, 269, 266, 261, 250, 235, 204, 165,  94;
420, 419, 416, 411, 400, 385, 354, 315, 244, 150;

Crossrefs

Mirror of triangle A212011. Column 1 is A066186. Right border is A138879.

Cf. A135010, A138121, A182703, A206563, A211980, A212000, A212010.

Sequence in context: A094885 A240199 A094728 * A365904 A370290 A275473

Adjacent sequences: A211998 A211999 A212000 * A212002 A212003 A212004

Keyword

nonn,tabl

Author

Omar E. Pol, Apr 26 2012

Status

approved