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A212001
Triangle read by rows: T(n,k) = sum of all parts of the last n-k+1 shells of n.
4
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.
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.
Sequence in context: A094885 A240199 A094728 * A365904 A370290 A275473
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Apr 26 2012
STATUS
approved