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A092321
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Sum of largest parts (counted with multiplicity) of all partitions of n.
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12
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0, 1, 4, 8, 17, 26, 49, 69, 115, 164, 249, 343, 513, 686, 974, 1314, 1806, 2382, 3232, 4208, 5597, 7244, 9456, 12118, 15687, 19899, 25422, 32079, 40589, 50796, 63805, 79303, 98817, 122179, 151145, 185820, 228598, 279476, 341807, 416051, 506205, 613244, 742720
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OFFSET
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0,3
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LINKS
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FORMULA
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G.f.: Sum_{n>=1} (n*x^n/(1-x^n))*Product_{k=1..n} 1/(1-x^k).
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EXAMPLE
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Partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4]; thus a(4) = 4*1 + 1*2 + 2*2 + 1*3 + 1*4 = 17.
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MAPLE
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b:= proc(n, i, t) option remember; `if`(n=0, [1, 0],
`if`(i<1, [0$2], b(n, i-1, t) +add((l->`if`(t, l,
l+[0, l[1]*i*j]))(b(n-i*j, i-1, true)), j=1..n/i)))
end:
a:= n-> b(n$2, false)[2]:
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MATHEMATICA
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f[n_] := Block[{c = 2n, k = 2, p = IntegerPartitions[n]}, m = Max @@@ p; l = Length[p]; While[k < l, c = c + m[[k]]*Count[p[[k]], m[[k]]]; k++ ]; If[n == 1, 1, c]]; Table[ f[n], {n, 41}] (* Robert G. Wilson v, Feb 18 2004, updated by Jean-François Alcover, Jan 29 2014 *)
nmax = 50; CoefficientList[Series[Sum[n*x^n/(1-x^n) * Product[1/(1 - x^k), {k, 1, n}], {n, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 06 2019 *)
Join[{0}, Table[Total[Flatten[First[Split[#]]&/@IntegerPartitions[n]]], {n, 50}]] (* Harvey P. Dale, Oct 29 2019 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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