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A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1. 53
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Omar E. Pol, Illustration of the seven regions of 5

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

EXAMPLE

From Omar E. Pol, Aug 13 2013: (Start)

Illustration of initial terms as number of regions:

.                                           _ _ _ _ _ _

.                                          |_ _ _      |

.                                          |_ _ _|_    |

.                                          |_ _    |   |

.                               _ _ _ _ _  |_ _|_ _|_  |

.                              |_ _ _    |           | |

.                     _ _ _ _  |_ _ _|_  |           | |

.                    |_ _    |         | |           | |

.             _ _ _  |_ _|_  |         | |           | |

.       _ _  |_ _  |       | |         | |           | |

.   _  |_  |     | |       | |         | |           | |

.  |_|   |_|     |_|       |_|         |_|           |_|

.

.   1     1       1         2           2             4

.

(End)

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 *)

CROSSREFS

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

Sequence in context: A036000 A002865 A085811 * A014810 A239835 A026929

Adjacent sequences:  A187216 A187217 A187218 * A187220 A187221 A187222

KEYWORD

nonn,less,changed

AUTHOR

Omar E. Pol, Dec 09 2011

EXTENSIONS

Better definition from Omar E. Pol, Sep 04 2013

STATUS

approved

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Last modified February 19 03:24 EST 2018. Contains 299330 sequences. (Running on oeis4.)