A092265 Sum of smallest parts of all partitions of n into distinct parts.

1, 2, 4, 5, 8, 10, 14, 16, 23, 26, 34, 40, 50, 58, 74, 83, 102, 120, 142, 164, 198, 226, 266, 308, 359, 412, 482, 548, 634, 730, 834, 950, 1094, 1240, 1416, 1609, 1826, 2068, 2350, 2648, 2994, 3382, 3806, 4280, 4826, 5408, 6070, 6806, 7619, 8522, 9534, 10632

Offset: 1

Author

Vladeta Jovovic, Feb 14 2004

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A046746, A005895, A006128, A092269, A092319, A092316, A034296, A237665.

Cf. A026832, A336902, A336903.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n,i+1)+b(n-i, i+1)))
    end:
a:= n-> add(j*b(n-j, j+1), j=1..n):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 03 2016

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i + 1] + b[n - i, i + 1]]]; a[n_] := Sum[j*b[n - j, j + 1], {j, 1, n}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jan 21 2017, after Alois P. Heinz *)

Formula

G.f.: Sum_{n >= 1} (-1 + Product_{k >= n} 1 + x^k).

G.f.: Sum_{n >= 1} n*x^n*Product_{k >= n+1} (1 + x^k). - Joerg Arndt, Jan 29 2011

G.f.: Sum_{k >= 1} x^(k*(k+1)/2)/(1 - x^k)/Product_{i = 1..k} (1 - x^i). - Vladeta Jovovic, Aug 10 2004

Conjecture: a(n) = A034296(n) + A237665(n+1). - George Beck, May 06 2017

a(n) ~ exp(Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 20 2018

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 25 2004