A182736 -id:A182736 - cp's OEIS frontend

A182742 Table of partitions that do not contain 1 as a part for even integers.

2, 4, 2, 3, 2, 2, 6, 3, 2, 2, 5, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 8, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 7, 3, 2, 2, 2, 2, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 5, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 3, 2, 2

Offset: 1

Omar E. Pol, Nov 30 2010, Dec 01 2010, Dec 04 2010

This array read by antidiagonals is the main table of the shell model of partitions for even integers. Here the last sections of all even numbers are superimposed as shells of an onion. In this way many bits of information are saved.
The table is the head of the last section of partitions of an even integer when it tends to be infinite. Row n lists the parts of the n-th partition that do not contains 1 as a part.
The shell model of partitions uses this table during the filling mechanism of the head of the last section of the next even integer k. For example, in a mechanical version, the head of the last section (as a mirror) pivoting from vertical to horizontal position. Then a copy of the partitions of the integer k, listed in this table, is transmitted (or reflected) at the head (or mirror) of the last section. Finally the head (or mirror) pivots back to return to its original vertical position. And so on for all even integers.
In another version, simply a copy of the partitions of the integer k, listed in the table, are placed above the partitions of the last odd number placed in the vertical plane structure.
It appears this table is useful to know the structure of the partitions of all even integers. The same applies for odd numbers in the table of A182743. Furthermore, both tables can be unified in a three-dimensional shell model.

			Array begins:
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 2, 2, 2, 2, 2, 2, 2, 2,
6, 2, 2, 2, 2, 2, 2, 2, 2,
5, 3, 2, 2, 2, 2, 2, 2,
4, 4, 2, 2, 2, 2, 2,
8, 2, 2, 2, 2, 2,
4, 3, 3, 2, 2,
7, 3, 2, 2,
6, 4, 2,
5, 5,
10,

Cf. A135010, A138121, A141285, A182730, A182736, A182740, A182743, A182746.

Column 1 give A182732. Column 2 give A182744.

A182994 Sum of all parts of the n-th subshell of the head of the last section of the set of partitions of any even integer >= 2n.

Original entry on oeis.org

2, 6, 16, 32, 64, 132, 224, 404, 704, 1156, 1880, 3060, 4748, 7396, 11346, 17054, 25454, 37706, 54980, 79756, 114702, 163394, 231288, 325408, 454138, 630542, 870504, 1194536, 1631196, 2216992, 2997542, 4036022, 5411108, 7223636

Offset: 1

Omar E. Pol, Feb 06 2011

nonn

The last section of the set of partitions of 2n contains n subshells.

Also first differences of A182736. - Omar E. Pol, Mar 03 2011

			a(5)=64 because the 5th subshell of the head of the last section of any even integer >= 10 looks like this:
(10 . . . . . . . . . )
( 5 . . . . 5 . . . . )
( 6 . . . . . 4 . . . )
( 7 . . . . . . 3 . . )
( 4 . . . 3 . . 3 . . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
There are 17 parts whose sum is 10+5+5+6+4+7+3+4+3+3+2+2+2+2+2+2+2 = 10*5 + 2*7 = 64, so a(5)=64.

Cf. A135010, A138880, A182736, A182742, A182982, A182992, A182995.

a(1) = 2. a(n) = A138880(2n) - A138880(2n-2), n >= 2.

More terms from Omar E. Pol, Mar 03 2011

A182737 Sum of parts in all partitions of 2n+1 that do not contain 1 as a part.

Original entry on oeis.org

0, 3, 10, 28, 72, 154, 312, 615, 1122, 1995, 3465, 5819, 9575, 15498, 24563, 38378, 59202, 90055, 135420, 201630, 297045, 433741, 628155, 902212, 1286348, 1821567, 2562126, 3581655, 4977867, 6879400, 9457318, 12936609, 17610320, 23863323, 32196090

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

Bisection (odd part) of A138880.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A135010, A138121, A138880, A182736, A182743.

b:= proc(n,i) option remember; local p,q;
      if n<0 then [0,0]
    elif n=0 then [1,0]
    elif i<2 then [0,0]
    else p, q:= b(n,i-1), b(n-i,i);
        [p[1]+q[1], p[2]+q[2]+q[1]*i]
      fi
    end:
a:= n-> b(2*n+1,2*n+1)[2]:
seq(a(n), n=0..34); # Alois P. Heinz, Dec 03 2010

b[n_, i_] := b[n, i] = Module[{p, q}, Which[n<0, {0, 0}, n == 0, {1, 0}, i<2, {0, 0}, True, {p, q} = {b[n, i-1], b[n-i, i]}; {p[[1]] + q[[1]], p[[2]] + q[[2]] + q[[1]]*i}]]; a[n_] := b[2*n + 1, 2*n+1][[2]]; Table[ a[n], {n, 0, 34}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = A005408(n)*A182747(n).

More terms from Alois P. Heinz, Dec 03 2010

A182812 Triangle read by rows in which row n lists the parts of the largest subshell of all partitions of 2n that do not contain 1 as a part.

Original entry on oeis.org

2, 4, 2, 6, 3, 3, 2, 2, 8, 4, 4, 5, 3, 2, 2, 2, 2, 10, 5, 5, 6, 4, 7, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 12, 6, 6, 7, 5, 8, 4, 4, 4, 4, 9, 3, 5, 4, 3, 6, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 14, 7, 7, 8, 6, 9, 5

Offset: 1

Omar E. Pol, Dec 04 2010

In the shell model of partitions the head of the last section of the set of partitions of 2n contains n subshells. The first n rows of this triangle represent these subsells.
This sequence contains the same elements of A182742 but in distinct order.
See A135010 and A138121 for more information.

			For n=1 the unique partition of 2n that does not contains 1 as part is 2, so row 1 has an element = 2.
For n=2 there are 2 partitions of 2n that do not contain 1 as part:
4 ............ or ....... 4 . . .
2 + 2 ........ or .......(2). 2 .
These partitions contain (2), the row n-1 of triangle, so
the parts of the largest subshell are 4, 2.
For n=3 there are 4 partitions of 2n that do not contain 1 as part:
6 ............ or ....... 6 . . . . .
3 + 3 ........ or ....... 3 . . 3 . .
4 + 2 ........ or .......(4). . . 2 .
2 + 2 + 2 .... or .......(2).(2). 2 .
These partitions contain (4) and (2),(2), the parts of rows < n of triangle, so the parts of the largest subshell are 6, 3, 3, 2, 2.
And so on.
Triangle begins:
2,
4, 2,
6, 3, 3, 2, 2,
8, 4, 4, 5, 3, 2, 2, 2, 2,
10, 5, 5, 6, 4, 7, 3, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2,

Cf. A135010, A138121, A182734, A182736, A182742, A182746, A182813.

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A182742 Table of partitions that do not contain 1 as a part for even integers.

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A182994 Sum of all parts of the n-th subshell of the head of the last section of the set of partitions of any even integer >= 2n.

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A182737 Sum of parts in all partitions of 2n+1 that do not contain 1 as a part.

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A182812 Triangle read by rows in which row n lists the parts of the largest subshell of all partitions of 2n that do not contain 1 as a part.

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