A046746 Sum of smallest parts of all partitions of n.

0, 1, 3, 5, 9, 12, 20, 25, 38, 49, 69, 87, 123, 152, 205, 260, 341, 425, 555, 687, 882, 1094, 1380, 1702, 2140, 2620, 3254, 3982, 4907, 5967, 7318, 8856, 10787, 13019, 15759, 18943, 22840, 27334, 32794, 39139, 46758, 55595, 66182, 78433, 93021, 109935, 129922

Offset

0,3

Comments

Also total number of largest parts in all partitions of n. - Vladeta Jovovic, Feb 16 2004

To see this, consider the properties of a partition related through conjugation, such as the total number of parts and the size of the largest parts. The sums over all of the partitions of n of these two properties are equal. The size of the smallest part and the number of largest parts are two such properties (this is immediate when looking at the Ferrers diagram). - Michael Donatz, Apr 17 2011

Starting with offset 1, = the partition triangle A026794 * [1, 2, 3, ...]. - Gary W. Adamson, Feb 13 2008

For n >= 1, a(n) = T(n+1,1) + T(n+2,2) + T(n+3,3)+ ... (sum along a falling diagonal) of the partition triangle A026794. - Bob Selcoe, Jun 22 2013

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000 (terms 0..10000 from Alois P. Heinz)

P. J. Grabner, A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.

Formula

G.f.: Sum_{k>=1} k*z^k/Product_{i>=0} (1-z^(k+i)). - Vladeta Jovovic, Jun 22 2003

G.f.: Sum_{k>=1} (-1 + 1/Product_{i>=0} (1-z^(k+i))). - Vladeta Jovovic, Jun 22 2003 [Cannot verify, Joerg Arndt, Apr 17 2011]

G.f.: Sum_{j>=1} (x^j/(1-x^j))/Product_{i=1..j} (1-x^i). - Vladeta Jovovic, Aug 11 2004 [Cannot verify, Joerg Arndt, Apr 17 2011]

G.f.: Sum_{k >= 1} (-1 + z^k/(1-z^k)(1-z^{k+1})(1-z^{k+2})...). - Don Knuth, Aug 08 2002 [Cannot verify, Joerg Arndt, Apr 17 2011]

G.f.: Sum_{n>=1} (x^n/(1-x^n)) / Product_{k=1..n} (1-x^k). - Joerg Arndt, May 26 2012

a(n) = A066186(n) - A066186(n-1) - A182709(n), n >= 1. - Omar E. Pol, Aug 01 2013

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n) * (1 + (23*Pi/(24*sqrt(6)) - sqrt(3/2)/Pi)/sqrt(n) + (1681*Pi^2/6912 - 23/16)/n). - Vaclav Kotesovec, Jul 06 2019

Example

For n = 4 the five partitions of 4 are 4, 2+2, 3+1, 2+1+1, 1+1+1+1, therefore the smallest parts of all partitions of 4 are 4, 2, 1, 1, 1 and the sum is 4+2+1+1+1 = 9, so a(4) = 9. - Omar E. Pol, Aug 02 2013

Maple

b:= proc(n, i) option remember;
      `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 28 2012

Mathematica

f[n_] := Plus @@ Min /@ IntegerPartitions@ n; Array[f, 45, 0] (* Robert G. Wilson v, Apr 12 2011 *)
b[n_, i_] := b[n, i] = If[n==i, n, 0] + If[i<1, 0, b[n, i-1] + If[n<i, 0, b[n-i, i]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 31 2015, after Alois P. Heinz *)

Prog

(PARI) N=66; z='z+O('z^N);  gf=sum(k=1, N, k * z^k / prod(j=k, N, 1-z^j ) ); concat([0], Vec(gf)) \\ Joerg Arndt, Apr 17 2011

Crossrefs

Cf. A006128, A026794, A336902, A336903.

Row sums of A026807.

Sequence in context: A247799 A265702 A190310 * A344715 A058599 A238662

Adjacent sequences: A046743 A046744 A046745 * A046747 A046748 A046749

Keyword

nonn,nice

Author

David W. Wilson

Status

approved