A212011 Triangle read by rows: T(n,k) = sum of all parts of the last k shells of n.

1, 3, 4, 5, 8, 9, 11, 16, 19, 20, 15, 26, 31, 34, 35, 31, 46, 57, 62, 65, 66, 39, 70, 85, 96, 101, 104, 105, 71, 110, 141, 156, 167, 172, 175, 176, 94, 165, 204, 235, 250, 261, 266, 269, 270, 150, 244, 315, 354, 385, 400, 411, 416, 419, 420, 196, 346

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(n-k).

T(n,k) = Sum_{j=n-k+1..n} A138879(j).

Example

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

Crossrefs

Mirror of triangle A212001. Column 1 is A138879. Right border is A066186.

Cf. A135010, A182703, A212000, A212010.

Sequence in context: A030310 A178837 A217288 * A207005 A316976 A256173

Adjacent sequences: A212008 A212009 A212010 * A212012 A212013 A212014

Keyword

nonn,tabl

Author

Omar E. Pol, Apr 26 2012

Status

approved