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

1,4

Links

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

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]

Example

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

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 *)

Crossrefs

Cf. A006141.

Sequence in context: A326461 A326547 A326686 * A073819 A190846 A214787

Adjacent sequences: A096400 A096401 A096402 * A096404 A096405 A096406

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 07 2004

Extensions

More terms from Emeric Deutsch, Jul 25 2005

Status

approved