login
This site is supported by donations to The OEIS Foundation.

 

Logo

110 people attended OEIS-50 (videos, suggestions); annual fundraising drive to start soon (donate); editors, please edit! (stack is over 300), your editing is more valuable than any donation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A026007 Expansion of prod(m>=1, (1+q^m)^m ); number of partitions of n into distinct parts, where n different parts of size n are available. 4
1, 1, 2, 5, 8, 16, 28, 49, 83, 142, 235, 385, 627, 1004, 1599, 2521, 3940, 6111, 9421, 14409, 21916, 33134, 49808, 74484, 110837, 164132, 241960, 355169, 519158, 755894, 1096411, 1584519, 2281926, 3275276, 4685731, 6682699, 9501979, 13471239, 19044780, 26850921, 37756561, 52955699 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Equals A000219: (1, 1, 3, 6, 13, 24, 48, 86,...) convolved with the aerated version of the latter: (1, 0, 1, 0, 3, 0, 6, 0, 13,...). - Gary W. Adamson, Jun 13 2009

LINKS

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

FORMULA

a(n) = 1/n*Sum_{k=1..n} A078306(k)*a(n-k). - Vladeta Jovovic, Nov 22 2002

G.f. Product_{m=1}^{infinity} (1+x^m)^m. Weighout transform of natural numbers (A000027). Euler transform of A026741. - Franklin T. Adams-Watters, Mar 16 2006

EXAMPLE

For n = 4, we have 8 partitions

01: [4]

02: [4']

03: [4'']

04: [4''']

05: [3, 1]

06: [3', 1]

07: [3'', 1]

08: [2, 2']

MAPLE

with(numtheory):

b:= proc(n) option remember;

      add((-1)^(n/d+1)*d^2, d=divisors(n))

    end:

a:= proc(n) option remember;

      `if`(n=0, 1, add(b(k)*a(n-k), k=1..n)/n)

    end:

seq(a(n), n=0..45);  # Alois P. Heinz, Aug 03 2013

MATHEMATICA

a[n_] := a[n] = 1/n*Sum[Sum[(-1)^(k/d+1)*d^2, {d, Divisors[k]}]*a[n-k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 41}] (* Jean-Fran├žois Alcover, Apr 17 2014, after Vladeta Jovovic *)

PROG

(PARI)

N=66; q='q+O('q^N);

gf= prod(n=1, N, (1+q^n)^n );

Vec(gf)

/* Joerg Arndt, Oct 06 2012 */

CROSSREFS

Cf. A000009, A000219, A000027, A026741.

Cf. A000219 [From Gary W. Adamson, Jun 13 2009]

Sequence in context: A137685 A169826 A093065 * A032233 A026530 A032254

Adjacent sequences:  A026004 A026005 A026006 * A026008 A026009 A026010

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 | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified October 31 00:18 EDT 2014. Contains 248844 sequences.