A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n.

1, 2, 3, 3, 5, 6, 6, 9, 11, 12, 8, 14, 17, 19, 20, 15, 23, 29, 32, 34, 35, 19, 34, 42, 48, 51, 53, 54, 32, 51, 66, 74, 80, 83, 85, 86, 42, 74, 93, 108, 116, 122, 125, 127, 128, 64, 106, 138, 157, 172, 180, 186, 189, 191, 192, 83, 147, 189, 221, 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.

Links

Table of n, a(n) for n=1..60.

Formula

T(n,k) = A006128(n) - A006128(n-k).

T(n,k) = Sum_{j=n-k+1..n} A138137(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 the first column equals the total number of parts 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
. ---------- ---------- ---------- ---------- ----------
.  8         14         17         19         20
.
So row 5 lists 8, 14, 17, 19, 20.
.
Triangle begins:
1;
2,    3;
3,    5,   6;
6,    9,  11,  12;
8,   14,  17,  19,  20;
15,  23,  29,  32,  34,  35;
19,  34,  42,  48,  51,  53,  54;
32,  51,  66,  74,  80,  83,  85,  86;
42,  74,  93, 108, 116, 122, 125, 127, 128;
64, 106, 138, 157, 172, 180, 186, 189, 191, 192;

Crossrefs

Mirror of triangle A212000. Column 1 is A138137. Right border is A006128.

Cf. A135010, A138121, A181187, A182703, A206563, A207031, A207032, A212001, A212011

Sequence in context: A178041 A181805 A369450 * A366418 A328745 A357134

Adjacent sequences: A212007 A212008 A212009 * A212011 A212012 A212013

Keyword

nonn,tabl

Author

Omar E. Pol, Apr 26 2012

Status

approved