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A066184
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Sum of the first moments of all partitions of n.
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2
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0, 1, 5, 13, 32, 61, 123, 208, 367, 590, 957, 1459, 2266, 3328, 4938, 7097, 10205, 14299, 20100, 27626, 38023, 51485, 69600, 92882, 123863, 163235, 214798, 280141, 364530, 470660, 606557, 776233, 991370, 1258827, 1594741, 2010142, 2528445
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OFFSET
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0,3
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COMMENTS
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The first element of each partition is given weight 1.
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
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FORMULA
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a(n) = 1/2*(A066183(n) + A066186(n)). - Vladeta Jovovic, Mar 23 2003
G.f.: Sum_{k>=1} x^k/(1 - x^k)^3 / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021
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EXAMPLE
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a(3)=13 because the first moments of all partitions of 3 are {3}.{1},{2,1}.{1,2} and {1,1,1}.{1,2,3}, resulting in 3,4,6; summing to 13.
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, [1, 0],
`if`(i<1, [0$2], `if`(i>n, b(n, i-1), b(n, i-1)+
(h-> h+[0, h[1]*i*(i+1)/2])(b(n-i, i)))))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50); # Alois P. Heinz, Jan 29 2014
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MATHEMATICA
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Table[ Plus@@ Map[ #.Range[ Length[ # ] ]&, IntegerPartitions[ n ] ], {n, 40} ]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, If[i > n, b[n, i - 1], b[n, i - 1] + Function[h, h + {0, h[[1]]*i*(i + 1)/2}][b[n - i, i]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)
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CROSSREFS
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Cf. A066185.
Sequence in context: A046789 A271902 A272539 * A231799 A146924 A342032
Adjacent sequences: A066181 A066182 A066183 * A066185 A066186 A066187
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KEYWORD
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easy,nonn
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AUTHOR
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Wouter Meeussen, Dec 15 2001
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STATUS
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approved
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