A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.

1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701, 35717

Offset: 1

Author

Omar E. Pol, Dec 09 2011

Keywords

• nonn

Comments

Essentially the same as A002865, but here a(1) = 1 not 0.

Also number of regions in the last section of the set of partitions of n.

Also number of partitions of n+k that are formed by k+1 sections, k >= 0 (Cf. A194799). - Omar E. Pol, Jan 30 2012

For the definition of region see A206437. - Omar E. Pol, Aug 13 2013

Partial sums give A000041, n >= 1. - Omar E. Pol, Sep 04 2013

Also the number of partitions of n with no parts greater than the number of ones. - Spencer Miller, Jan 28 2023

Examples

			From _Omar E. Pol_, Aug 13 2013: (Start)
Illustration of initial terms as number of regions:
.                                           _ _ _ _ _ _
.                                          |_ _ _      |
.                                          |_ _ _|_    |
.                                          |_ _    |   |
.                               _ _ _ _ _  |_ _|_ _|_  |
.                              |_ _ _    |           | |
.                     _ _ _ _  |_ _ _|_  |           | |
.                    |_ _    |         | |           | |
.             _ _ _  |_ _|_  |         | |           | |
.       _ _  |_ _  |       | |         | |           | |
.   _  |_  |     | |       | |         | |           | |
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1     1       1         2           2             4
.
(End)
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
• Omar E. Pol, Illustration of the seven regions of 5

Crossrefs

Cf. A000041, A002865, A083751, A135010, A186114, A193870, A206437, A225600, A225610.

Programs

Mathematica

Join[{1}, Drop[CoefficientList[Series[1 / Product[(1 - x^k)^1, {k, 2, 50}], {x, 0, 50}], x], 2]] (* Vincenzo Librandi, Feb 15 2018 *)
A187219[nmax_]:=Join[{1},Differences[PartitionsP[Range[nmax]]]];
A187219[100] (* Paolo Xausa, Feb 17 2023 *)

Formula

a(n) = A083751(n) + 1. - Omar E. Pol, Mar 04 2012

a(n) = A002865(n), if n >= 2. - Omar E. Pol, Aug 13 2013

Extensions

Better definition from Omar E. Pol, Sep 04 2013