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A212011
Triangle read by rows: T(n,k) = sum of all parts of the last k shells of n.
4
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.
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.
Sequence in context: A030310 A178837 A217288 * A207005 A316976 A256173
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Apr 26 2012
STATUS
approved