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A046746 Sum of smallest parts of all partitions of n.

%I A046746 #102 Aug 24 2025 17:57:53
%S A046746 0,1,3,5,9,12,20,25,38,49,69,87,123,152,205,260,341,425,555,687,882,
%T A046746 1094,1380,1702,2140,2620,3254,3982,4907,5967,7318,8856,10787,13019,
%U A046746 15759,18943,22840,27334,32794,39139,46758,55595,66182,78433,93021,109935,129922
%N A046746 Sum of smallest parts of all partitions of n.
%C A046746 Also total number of largest parts in all partitions of n. - _Vladeta Jovovic_, Feb 16 2004
%C A046746 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
%C A046746 Starting with offset 1, = the partition triangle A026794 * [1, 2, 3, ...]. - _Gary W. Adamson_, Feb 13 2008
%C A046746 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
%H A046746 Vaclav Kotesovec, <a href="/A046746/b046746.txt">Table of n, a(n) for n = 0..20000</a> (terms 0..10000 from Alois P. Heinz)
%H A046746 P. J. Grabner, A. Knopfmacher, <a href="https://www.math.tugraz.at/fosp/pdfs/tugraz_0087.pdf">Analysis of some new partition statistics</a>, Ramanujan J., 12, 2006, 439-454.
%F A046746 G.f.: Sum_{k>=1} k*z^k/Product_{i>=0} (1-z^(k+i)). - _Vladeta Jovovic_, Jun 22 2003
%F A046746 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]
%F A046746 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]
%F A046746 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]
%F A046746 G.f.: Sum_{n>=1} (x^n/(1-x^n)) / Product_{k=1..n} (1-x^k). - _Joerg Arndt_, May 26 2012
%F A046746 a(n) = A066186(n) - A066186(n-1) - A182709(n), n >= 1. - _Omar E. Pol_, Aug 01 2013
%F A046746 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
%e A046746 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
%p A046746 b:= proc(n, i) option remember;
%p A046746       `if`(n=i, n, 0) +`if`(i<1, 0, b(n, i-1) +`if`(n<i, 0, b(n-i, i)))
%p A046746     end:
%p A046746 a:= n-> b(n, n):
%p A046746 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 28 2012
%t A046746 f[n_] := Plus @@ Min /@ IntegerPartitions@ n; Array[f, 45, 0] (* _Robert G. Wilson v_, Apr 12 2011 *)
%t A046746 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_ *)
%t A046746 Join[{0},Table[Total[IntegerPartitions[n][[;;,-1]]],{n,50}]] (* _Harvey P. Dale_, Aug 24 2025 *)
%o A046746 (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
%Y A046746 Cf. A006128, A026794, A336902, A336903.
%Y A046746 Row sums of A026807.
%K A046746 nonn,nice,changed
%O A046746 0,3
%A A046746 _David W. Wilson_