The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A000041 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A000041

a(n) is the number of partitions of n (the partition numbers).
(history; published version)

#1108 by Michael De Vlieger at Sun Aug 25 09:56:28 EDT 2024

STATUS

reviewed

approved

#1107 by Michel Marcus at Sun Aug 25 05:07:34 EDT 2024

STATUS

proposed

reviewed

#1106 by Joerg Arndt at Sun Aug 25 04:42:41 EDT 2024

STATUS

editing

proposed

#1105 by Joerg Arndt at Sun Aug 25 04:42:36 EDT 2024

FORMULA

a(n) = Sum_{c=0..floor(sqrt(n))} Sum_{i=0...n-c^2} A026820(i,c)*A026820(n-c^2-i,c). - David García Herrero, Aug 20 2024

STATUS

proposed

editing

#1104 by Alois P. Heinz at Wed Aug 21 14:07:58 EDT 2024

STATUS

editing

proposed

#1103 by Alois P. Heinz at Wed Aug 21 14:06:23 EDT 2024

COMMENTS

a(n) corresponds to is also the number of partitions of n*(n+3)/2 into n distinct parts. - David García Herrero, Aug 20 2024

FORMULA

For n>0, a(n) = Sum_{c=10..floor(sqrt(n))} Sum_{i=0...n-c^2} pA026820(i,c)*pA026820(n-c^2-i,c), where p(i,c) is the number of partitions of i into at most c parts. - David García Herrero, Aug 20 2024

CROSSREFS

Cf. A000009, A000079, A000203, A001318, A008284, A026820, A065446, A078506, A113685, A132311, A000248.

Discussion

Wed Aug 21

14:07

Alois P. Heinz: ...

#1102 by Alois P. Heinz at Tue Aug 20 22:16:48 EDT 2024

STATUS

proposed

editing

Discussion

Tue Aug 20

22:17

Alois P. Heinz: your proposed formula is too complicated ...

#1101 by David García Herrero at Tue Aug 20 20:55:51 EDT 2024

STATUS

editing

proposed

Discussion

Tue Aug 20

22:16

Alois P. Heinz: A026820(n,k) is number of partitions of n into at most k parts. 
And we already have the comment: a(n) = A026820(n, n) ...

#1100 by David García Herrero at Tue Aug 20 20:46:45 EDT 2024

FORMULA

For n>0, a(n) = Sum_{c=1..floor(sqrt(n))} Sum_{i=0...n-c^2} p(i,c)*p(n-c^2-i,c), where p(i,c) is the number of partitions of i into at most c parts. - David García Herrero, Aug 20 2024

Discussion

Tue Aug 20

20:53

David García Herrero: N in N(N+3)/2 is a typo. It is actually n: n(n+3)/2, it is already corrected

#1099 by David García Herrero at Tue Aug 20 20:03:45 EDT 2024

COMMENTS

a(n) corresponds to the number of partitions of Nn(Nn+3)/2 into N n distinct parts. - David García Herrero, Aug 20 2024

Discussion

Tue Aug 20

20:45

David García Herrero: By convention, p(0)=1 therefore a(0)=1 without the need for operations. The formula is deduced from the characteristics of the conjugate partitions that do not exist in zero; It is logical that in that case the formula is not applicable. 
This does not make it incorrect or not applicable to other natural numbers. You can specify "for n>0 a(n)= ...etc