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

Author

Clark Kimberling

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A025147, A025149.

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

Programs

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 `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 *)

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

Extensions

More terms from Emeric Deutsch, Apr 17 2006