A207032 -id:A207032 - cp's OEIS frontend

A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.

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

Omar E. Pol, Apr 26 2012

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.

			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;

Mirror of triangle A212010. Column 1 is A006128. Right border gives A138137.

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

T(n,k) = A006128(n) - A006128(k-1).

T(n,k) = Sum_{j=k..n} A138137(j).

A208476 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

Original entry on oeis.org

1, 1, 2, 5, 0, 3, 3, 8, 0, 4, 13, 2, 8, 0, 5, 13, 18, 6, 10, 0, 6

Offset: 1

Omar E. Pol, Feb 28 2012

Essentially this sequence is related to A206562 in the same way as A207032 is related to A207031 and also in the same way as A206563 is related to A181187. See the calculation in the example section of A206563.

			Triangle begins:
1;
1,   2;
5,   0,  3;
3,   8,  0,  4;
13,  2,  8,  0,  5;
13, 18,  6, 10,  0,  6;

Right border is A000027.

Cf. A135010, A182703, A206562, A207031, A207032, A207383, A208475.

A210955 Triangle read by rows: T(n,k) = total number of parts <= k in the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 5, 5, 6, 5, 6, 7, 7, 8, 7, 11, 13, 14, 14, 15, 11, 14, 16, 17, 18, 18, 19, 15, 23, 26, 29, 30, 31, 31, 32, 22, 29, 35, 37, 39, 40, 41, 41, 42, 30, 45, 51, 56, 59, 61, 62, 63, 63, 64, 42, 57, 67, 72, 76, 78, 80, 81, 82, 82, 83

Offset: 1

Omar E. Pol, May 01 2012

Row n lists the partial sums of row n of triangle A182703.

			1,
1,   2,
2,   2,  3,
3,   5,  5,  6,
5,   6,  7,  7,  8,
7,  11, 13, 14, 14, 15,
11, 14, 16, 17, 18, 18, 19,
15, 23, 26, 29, 30, 31, 31, 32,
22, 29, 35, 37, 39, 40, 41, 41, 42;

Column 1 is A000041. Right border gives A138137.

Cf. A135010, A138121, A182703, A206437, A206562, A207031, A207032, A207383, A208476, A210947, A210956.

T(n,k) = Sum_{j=1..k} A182703(n,j).

More terms from Alois P. Heinz, May 25 2013

cp's OEIS Frontend

A212000 Triangle read by rows: T(n,k) = total number of parts in the last n-k+1 shells of n.

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A208476 Triangle read by rows: T(n,k) = total sum of odd/even parts >= k in the last section of the set of partitions of n, if k is odd/even.

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A210955 Triangle read by rows: T(n,k) = total number of parts <= k in the last section of the set of partitions of n.

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