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A046746
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Sum of smallest parts of all partitions of n.
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46
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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
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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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.
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FORMULA
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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(x^j/(1-x^j)/Product(1-x^i, i=1..j), j=1..infinity). - 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) / prod(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
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A006128, A026794.
Row sums of A026807.
Sequence in context: A247799 A265702 A190310 * A058599 A238662 A059093
Adjacent sequences: A046743 A046744 A046745 * A046747 A046748 A046749
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KEYWORD
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nonn,nice
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AUTHOR
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David W. Wilson
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STATUS
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approved
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