login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A047967 Number of partitions of n with some part repeated. 30
0, 0, 1, 1, 3, 4, 7, 10, 16, 22, 32, 44, 62, 83, 113, 149, 199, 259, 339, 436, 563, 716, 913, 1151, 1453, 1816, 2271, 2818, 3496, 4309, 5308, 6502, 7959, 9695, 11798, 14298, 17309, 20877, 25151, 30203, 36225, 43323, 51748, 61651, 73359, 87086, 103254, 122164 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Also number of partitions of n with at least one even part. - Vladeta Jovovic, Sep 10 2003. Example: a(5)=4 because we have [4,1], [3,2], [2,2,1] and [2,1,1,1] ([5], [3,1,1] and [1,1,1,1,1] do not qualify). - Emeric Deutsch, Mar 30 2006

Also number of partitions of n (where it is assumed that the least part is 0) such that at least one difference is at least two. Example: a(5)=4 because we have [5,0], [4,1,0], [3,2,0] and [3,1,1,0] ([2,2,1,0], [2,1,1,1,0] and [1,1,1,1,1,0] do not qualify). - Emeric Deutsch, Mar 30 2006

The Heinz numbers of these partitions (with some part repeated) are given by A013929. Equivalent to Vladeta Jovovic's comment, a(n) is also the number of integer partitions whose product of parts is even. The Heinz numbers of these latter partitions are given by A324929. - Gus Wiseman, Mar 23 2019

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

H. Bottomley, Illustration for A000009, A000041, A047967

FORMULA

a(n) = A000041(n) - A000009(n).

G.f.: sum(x^(2*k)*product(1+x^j, j=k+1..infinity)/product(1-x^j, j=1..k), k=1..infinity) = sum(x^(2k)/(product(1-x^j, j=1..2*k)*product(1-x^(2*j+1), j=k..infinity) ), k=1..infinity). - Emeric Deutsch, Mar 30 2006

G.f.: 1/P(x) - P(x^2)/P(x) where P(x)=prod(k>=1, 1-x^k ). - Joerg Arndt, Jun 21 2011

EXAMPLE

a(5) = 4 because we have [3,1,1], [2,2,1], [2,1,1,1] and [1,1,1,1,1] ([5], [4,1] and [3,2] do not qualify).

MAPLE

g:=sum(x^(2*k)*product(1+x^j, j=k+1..70)/product(1-x^j, j=1..k), k=1..40): gser:=series(g, x=0, 50): seq(coeff(gser, x, n), n=0..44); # Emeric Deutsch, Mar 30 2006

MATHEMATICA

Clear[fQ, fP, lst, n]; fQ[n_]:=PartitionsQ[n]; fP[n_]:=PartitionsP[n]; lst={}; Do[AppendTo[lst, fP[n]-fQ[n]], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 19 2009 *)

Table[PartitionsP[n]-PartitionsQ[n], {n, 0, 50}] (* Harvey P. Dale, Jan 17 2019 *)

PROG

(PARI)  x='x+O('x^66); concat([0, 0], Vec(1/eta(x)-eta(x^2)/eta(x))) \\ Joerg Arndt, Jun 21 2011

CROSSREFS

Cf. A038348, A261982.

Column k=1 of A320264.

Cf. A324847, A324929, A324966, A324967.

Sequence in context: A004397 A324368 A241654 * A282893 A256912 A134591

Adjacent sequences:  A047964 A047965 A047966 * A047968 A047969 A047970

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 9 18:32 EDT 2020. Contains 336326 sequences. (Running on oeis4.)