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A188814
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Sum of the "complements" of the integer partitions of n.
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3
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0, 0, 0, 1, 4, 12, 27, 57, 107, 192, 327, 538, 855, 1329, 2018, 3003, 4402, 6349, 9045, 12720, 17713, 24395, 33335, 45118, 60655, 80888, 107242, 141177, 184905, 240679, 311850, 401860, 515725, 658630, 838006, 1061561, 1340193, 1685271, 2112576, 2638727
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OFFSET
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0,5
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COMMENTS
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Consider the m x k rectangle corresponding to an integer partition p of n, where m is the greatest part of p and k is the number of parts of p. Fit the Ferrers diagram of p inside its corresponding rectangle. a(n) is the number of empty spaces in all such rectangles over all partitions of n.
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REFERENCES
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Sriram Pemmaraju and Steven Skiena, Computational Discrete Mathematics, Cambridge, 2003, page 145.
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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) = Sum_{k>0} k*A268192(n,k). - Alois P. Heinz, Feb 12 2016
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EXAMPLE
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a(4) = 4 because the partitions 4, 2+2, 1+1+1+1 have no empty spaces while the partitions 3+1 and 2+1+1 each have two.
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MAPLE
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b:= proc(n, i) option remember; local f, g;
if n=0 or i=1 then [1, n]
elif i<1 then [0, 0]
else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
[f[1]+g[1], f[2]+g[2]+g[1]]
fi
end:
a:= n-> add(add(i, i=b(n-j, min(j, n-j)))*j, j=1..n) -n*b(n, n)[1]:
seq(a(n), n=0..40); # Alois P. Heinz, Apr 22 2011, Apr 11 2012
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MATHEMATICA
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f[list_]:= Total[Select[Reverse[Table[Max[list]-list[[i]], {i, 1, Length[list]}]], #>0&]];
Table[Total[Map[f, IntegerPartitions[n]]], {n, 0, 30}]
(* second program: *)
b[n_, i_] := b[n, i] = Module[{f, g}, If [n==0 || i==1, {1, n}, If[i<1, {0, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]];
a[n_] := Sum[Sum[i, {i, b[n-j, Min[j, n-j]]}]*j, {j, 1, n}]-n*b[n, n][[1]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)
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CROSSREFS
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Cf. A066186, A182094, A268192.
Sequence in context: A062479 A327032 A007009 * A104384 A013697 A306055
Adjacent sequences: A188811 A188812 A188813 * A188815 A188816 A188817
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KEYWORD
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nonn
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AUTHOR
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Geoffrey Critzer, Apr 22 2011
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STATUS
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approved
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