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A046746
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Sum of smallest parts of all partitions of n.
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30
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Also total number of largest parts in all partitions of n. - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 (the proof falls out almost immediately if you look at a Ferrers diagram) [Michael Donatz, Apr 17 2011]
Starting with offset 1, = the partition triangle A026794 * [1, 2, 3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 13 2008
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FORMULA
| G.f.: Sum_{k>=1} k*z^k/Product_{i>=0} (1-z^(k+i)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 22 2003
G.f.: Sum_{k>=1} (-1+1/Product_{i>=0} (1-z^(k+i))) - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 (vladeta(AT)eunet.rs), 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})...). - D. E. Knuth, Aug 08, 2002 [Cannot verify, Joerg Arndt, Apr 17 2011]
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MATHEMATICA
| f[n_] := Plus @@ Min /@ IntegerPartitions@ n; Array[f, 45, 0] (* Robert G. Wilson v, Apr 12 2011 *)
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PROG
| (Pari) N=66; z='z+O('z^N); /* that many terms */
s=sum(k=1, N, k * z^k / prod(j=k, N, 1-z^j ) ); /* g.f. */
Vec(s) /* show terms */ /* Joerg Arndt, Apr 17 2011 */
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CROSSREFS
| Cf. A006128, A026794.
Row sums of A026807.
Sequence in context: A060419 A005766 A190310 * A058599 A059093 A084593
Adjacent sequences: A046743 A046744 A046745 * A046747 A046748 A046749
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KEYWORD
| nonn
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AUTHOR
| Dave Wilson
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