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A046746
Sum of smallest parts of all partitions of n.
48
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
Row sums of A026807.
Sequence in context: A247799 A265702 A190310 * A344715 A058599 A238662
KEYWORD
nonn,nice
STATUS
approved