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A212010 Triangle read by rows: T(n,k) = total number of parts in the last k shells of n. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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: A242642 A178041 A181805 * A328745 A003967 A099209

Adjacent sequences:  A212007 A212008 A212009 * A212011 A212012 A212013

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Apr 26 2012

STATUS

approved

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Last modified March 7 19:13 EST 2021. Contains 341928 sequences. (Running on oeis4.)