A096403 Number of partitions of n in which number of least parts is equal to least part.

1, 0, 1, 2, 2, 2, 5, 5, 9, 10, 14, 17, 26, 30, 41, 52, 67, 81, 108, 129, 168, 204, 257, 311, 393, 470, 584, 705, 865, 1036, 1270, 1514, 1838, 2192, 2639, 3137, 3767, 4455, 5321, 6287, 7469, 8794, 10419, 12230, 14427, 16904, 19863, 23210, 27207, 31701, 37039

Offset: 1

Author

Vladeta Jovovic, Aug 07 2004

Keywords

• easy
• nonn

Examples

			a(7)=5 because we have 61, 421, 331, 322 and 2221.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006141.

Programs

Maple

G:=sum((x^(m^2)-x^(m*(m+1)))/product(1-x^i,i=m..80),m=1..80): Gser:=series(G,x=0,70): seq(coeff(Gser,x^n),n=1..60); # Emeric Deutsch, Jul 25 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> add(b(n-j^2, j+1), j=1..isqrt(n)):
seq(a(n), n=1..55);  # Alois P. Heinz, Jan 25 2021

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := Sum[b[n - j^2, j+1], {j, 1, Sqrt[n]}];
Array[a, 55] (* Jean-François Alcover, Mar 01 2021, after Alois P. Heinz *)

Formula

G.f.: Sum((x^(m^2)-x^(m*(m+1)))/Product(1-x^i, i=m..infinity), m=1..infinity).

G.f.: sum(n>=1, x^(n^2) / prod(k>=n+1,1-x^k)). [Joerg Arndt, Mar 23 2011]

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). - Vaclav Kotesovec, Jun 15 2025

Extensions

More terms from Emeric Deutsch, Jul 25 2005