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A340493 Sequence whose partial sums give A340492. 1
1, 3, 8, 23, 49, 125, 241, 540, 1020, 2064, 3710, 7231, 12457, 22883, 39053, 68596, 113751, 194865, 315910, 526019, 840939, 1363524, 2144528, 3419185, 5291079, 8277252, 12668264, 19497436, 29459144, 44762200, 66847518, 100267761, 148318881, 219818270, 322056529, 472600353 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

In other words: 1 together with the first differences of A340492.

Conjecture: a(n) is the size of the n-th section of a table of correspondence between divisors and partitions.

LINKS

Table of n, a(n) for n=1..36.

FORMULA

a(1) = 1.

a(n) = A000041(n)*A000070(n-1) - A000041(n-1)*A000070(n-2), n >= 2.

EXAMPLE

Illustration of initial terms:

       A000070:  1 2   4     7        12            19                    30

A000041         _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

     1         |_| |   |     |         |             |                     |

     2         |_ _|   |     |         |             |                     |

     3         |_ _ _ _|     |         |             |                     |

               |             |         |             |                     |

     5         |_ _ _ _ _ _ _|         |             |                     |

               |                       |             |                     |

     7         |_ _ _ _ _ _ _ _ _ _ _ _|             |                     |

               |                                     |                     |

               |                                     |                     |

               |                                     |                     |

    11         |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|                     |

               |                                                           |

               |                                                           |

               |                                                           |

    15         |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _|

...

a(n) is the area (or the number of cells) in the n-th region (or section) of the diagram.

For n = 3 the third region of the diagram contains 8 cells, so a(3) = 8.

For n = 7 the seventh region of the diagram contains 241 cells, so a(7) = 241.

MATHEMATICA

a[n_] := PartitionsP[n]*Count[Flatten[IntegerPartitions[n]], 1] - PartitionsP[n - 1]*Count[Flatten[IntegerPartitions[n - 1]], 1]; Table[a[n], {n, 1, 36}] (* Robert P. P. McKone, Jan 28 2021 *)

PROG

(PARI) f(n) = numbpart(n)*sum(k=0, n-1, numbpart(k)); \\ A340492

a(n) = if (n==1, 1, f(n)-f(n-1)); \\ Michel Marcus, Jan 28 2021

CROSSREFS

Partial sums give A340492.

Cf. A000041, A000070, A090982, A336811.

Sequence in context: A073051 A183930 A183922 * A296286 A068602 A255834

Adjacent sequences:  A340490 A340491 A340492 * A340494 A340495 A340496

KEYWORD

nonn

AUTHOR

Omar E. Pol, Jan 10 2021

STATUS

approved

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Last modified July 25 16:39 EDT 2021. Contains 346291 sequences. (Running on oeis4.)