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A212000
Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.
4
1, 3, 2, 6, 5, 3, 12, 11, 9, 6, 20, 19, 17, 14, 8, 35, 34, 32, 29, 23, 15, 54, 53, 51, 48, 42, 34, 19, 86, 85, 83, 80, 74, 66, 51, 32, 128, 127, 125, 122, 116, 108, 93, 74, 42, 192, 191, 189, 186, 180, 172, 157, 138, 106, 64, 275, 274, 272, 269, 263, 255, 240
OFFSET
1,2
COMMENTS
The set of partitions of n contains n shells (see A135010). Let m and n be two positive integers such that m <= n. It appears that in any set formed by m connected shells, or m disconnected shells, or a mixture of both, the sum of all parts of the j-th column equals the total number of parts >= j in the same set (see example). More generally it appears that any of these sets has the same properties mentioned in A206563 and A207031.
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) = A006128(n) - A006128(k-1).
T(n,k) = Sum_{j=k..n} A138137(j).
EXAMPLE
For n = 5 the illustration shows five sets containing the last n-k+1 shells of 5 and below we can see that the sum of all parts of the first column equals the total number of parts in 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
. ---------- ---------- ---------- ---------- ----------
. 20 19 17 14 8
.
So row 5 lists 20, 19, 17, 14, 8.
.
Triangle begins:
1;
3, 2;
6, 5, 3;
12, 11, 9, 6;
20, 19, 17, 14, 8;
35, 34, 32, 29, 23, 15;
54, 53, 51, 48, 42, 34, 19;
86, 85, 83, 80, 74, 66, 51, 32;
128, 127, 125, 122, 116, 108, 93, 74, 42;
192, 191, 189, 186, 180, 172, 157, 138, 106, 64;
CROSSREFS
Mirror of triangle A212010. Column 1 is A006128. Right border gives A138137.
Sequence in context: A049777 A193999 A210971 * A058401 A244426 A214417
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Apr 26 2012
STATUS
approved