A026824 Number of partitions of n into distinct parts, the least being 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

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 occur 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

For n > 3, a(n) is the Euler transform of [0,0,0,1,1,1,1] joined with the period 2 sequence [0,1, ...]. - Georg Fischer, Aug 18 2020

Links

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

Formula

From Emeric Deutsch, Apr 17 2006: (Start)

G.f.: (x^3)*Product_{j=4..infinity} (1+x^j).

G.f.: Sum_{k=1..infinity} x^(k*(k+5)/2)/(Product_{j=1..k-1} (1-x^j)). (End)

a(n) = A025149(n-3), n>3. - 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-François 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 *)
Join[{0}, Table[Count[Last /@ Select[IntegerPartitions@n, DeleteDuplicates[#] == # &], 3], {n, 1, 66}]] (* Robert Price, Jun 13 2020 *)

Crossrefs

Cf. A025147, A025149.

Cf. A096765 (least=1), A096749 (2), A022825 (4), A022826 (5), A022827 (6), A022828 (7), A022829 (8), A022830 (9), A022831 (10).

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