A144300 -id:A144300 - cp's OEIS frontend

A220486 a(n) = n(p(n)-d(n)): sum of all of parts of all partitions of n with at least one distinct part.

0, 0, 3, 8, 25, 42, 91, 144, 243, 380, 594, 852, 1287, 1834, 2580, 3616, 5015, 6822, 9272, 12420, 16548, 21956, 28819, 37608, 48875, 63232, 81162, 103936, 132327, 167880, 212040, 266976, 334587, 418404, 520765, 646848, 800495, 988418, 1216059, 1493200

Offset: 1

Omar E. Pol, Jan 18 2013

			For n = 6
-----------------------------------------------------
Partitions of 6            Value
-----------------------------------------------------
6 .......................... 0  (all parts are equal)
5 + 1 ...................... 6
4 + 2 ...................... 6
4 + 1 + 1 .................. 6
3 + 3 ...................... 0  (all parts are equal)
3 + 2 + 1 .................. 6
3 + 1 + 1 + 1 .............. 6
2 + 2 + 2 .................. 0  (all parts are equal)
2 + 2 + 1 + 1 .............. 6
2 + 1 + 1 + 1 + 1 .......... 6
1 + 1 + 1 + 1 + 1 + 1 ...... 0  (all parts are equal)
-----------------------------------------------------
The sum of the values is    42
On the other hand p(6) = A000041(6) = 11 and d(6) = A000005(6) = 4, so a(6) = 6*(p(6) - d(6)) = 6*(11 - 4) = 6*7 = 42.

Cf. A000005, A000041, A038040, A066186, A144300, A220477.

a(n) = n*(A000041(n) - A000005(n)) = A066186(n) - A038040(n) = n*A144300(n).

A361392 Number of integer partitions of n whose first differences have mean -1.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 3, 2, 5, 4, 8, 7, 12, 12, 19, 19, 29, 31, 43, 48, 65, 73, 97, 110, 142, 164, 208, 240, 301, 350, 432, 504, 617, 719, 874, 1019, 1228, 1434, 1717, 2001, 2385, 2778, 3292, 3831, 4522, 5252, 6177, 7164, 8392, 9722, 11352, 13125, 15283, 17643

Offset: 0

Gus Wiseman, Mar 13 2023

nonn

These are partitions where the first part minus the last part is the number of parts minus 1.

			The a(3) = 1 through a(11) = 8 partitions:
  (21)  .  (32)   (321)  (43)    (422)   (54)     (442)    (65)
           (311)         (331)   (4211)  (432)    (4321)   (533)
                         (4111)          (4221)   (4411)   (4331)
                                         (4311)   (52111)  (4421)
                                         (51111)           (5222)
                                                           (52211)
                                                           (53111)
                                                           (611111)
For example, the partition y = (4,2,2,1) has first differences (-2,0,-1), with mean -1, so y is counted under a(9).

For mean 0 we have A032741.
The 0-appended version is A047993.
For any negative mean we have A144300.
A000041 counts integer partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean.
A067538 counts partitions with integer mean, ranks A316413.
A326567/A326568 gives mean of prime indices, conjugate A326839/A326840.
A360614/A360615 gives mean of 0-appended first differences of prime indices.
Cf. A102627, A237363, A237832, A348551, A360688.

Table[Length[Select[IntegerPartitions[n],Mean[Differences[#]]==-1&]],{n,0,30}]

cp's OEIS Frontend

A220486 a(n) = n(p(n)-d(n)): sum of all of parts of all partitions of n with at least one distinct part.

Views

Author

Keywords

Examples

Crossrefs

Formula

A361392 Number of integer partitions of n whose first differences have mean -1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica