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

Author

Omar E. Pol, Apr 26 2012

Keywords

• nonn
• tabl

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.

Examples

			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.

Formula

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

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