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!)
A026824 Number of partitions of n into distinct parts, the least being 3. 3
0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 12, 15, 17, 20, 23, 27, 31, 36, 41, 47, 55, 62, 71, 81, 93, 105, 120, 135, 154, 174, 197, 221, 251, 281, 317, 356, 400, 447, 502, 561, 628, 701, 782, 871, 972, 1081, 1202, 1336, 1483, 1645, 1825, 2021, 2237, 2476 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Also number of partitions of n such that if k is the largest part, then k occurs exactly 3 times and each of the numbers 1,2,...,k-1 occurs at least once (these are the conjugates of the partitions described in the definition). Example: a(14)=3 because we have [3,3,3,2,2,1],[3,3,3,2,1,1,1] and [2,2,2,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Apr 17 2006

LINKS

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

FORMULA

G.f.: (x^3)product(1+x^j, j=4..infinity). G.f.: sum(x^(k(k+5)/2)/product(1-x^j, j=1..k-1), k=1..infinity). - Emeric Deutsch, Apr 17 2006

a(n) = A025149(n-3), n>3. - R. J. Mathar, Jul 31 2008

G.f.: x^3*product_{j=4..infinity} (1+x^j). - R. J. Mathar, Jul 31 2008

a(n) ~ exp(Pi*sqrt(n/3)) / (32*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Oct 30 2015

EXAMPLE

a(14) = 3 because we have [11,3], [7,4,3] and [6,5,3].

MAPLE

g:=x^3*product(1+x^j, j=4..80): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=1..59); # Emeric Deutsch, Apr 17 2006

# second Maple program:

b:= proc(n, i) option remember;

      `if`(n=0, 1, `if`((i-3)*(i+4)/2<n, 0,

       add(b(n-i*j, i-1), j=0..min(1, n/i))))

    end:

a:= n-> `if`(n<3, 0, b(n-3$2)):

seq(a(n), n=0..60);  # Alois P. Heinz, Feb 07 2014

MATHEMATICA

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[(i-3)(i+4)/2 < n, 0, Sum[b[n-i*j, i-1], {j, 0, Min[1, n/i]}]]]; a[n_] := If[n<3, 0, b[n-3, n-3]]; Table[a[n], {n, 0, 60}] (* Jean-Fran├žois Alcover, May 13 2015, after Alois P. Heinz *)

nmax = 100; CoefficientList[Series[x^3/((1+x)*(1+x^2)*(1+x^3)) * Product[1+x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 30 2015 *)

CROSSREFS

Cf. A025147.

Sequence in context: A029026 A003106 A185228 * A025149 A026799 A185326

Adjacent sequences:  A026821 A026822 A026823 * A026825 A026826 A026827

KEYWORD

nonn

AUTHOR

Clark Kimberling

EXTENSIONS

More terms from Emeric Deutsch, Apr 17 2006

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 March 29 14:48 EDT 2020. Contains 333107 sequences. (Running on oeis4.)