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A182099 Total area of the largest inscribed rectangles of all integer partitions of n.

%I A182099 #20 Dec 06 2020 09:07:09
%S A182099 0,1,4,8,18,29,54,82,136,202,309,441,658,915,1303,1790,2479,3337,4541,
%T A182099 6022,8045,10554,13876,17996,23409,30055,38634,49208,62650,79116,
%U A182099 99898,125213,156848,195339,242964,300707,371770,457493,562292,688451,841707,1025484
%N A182099 Total area of the largest inscribed rectangles of all integer partitions of n.
%C A182099 a(n) >= A000041(n)*A061017(n) for n>0 because the least largest inscribed rectangle of any integer partition of n is A061017(n) and A000041(n) is the number of partitions of n.
%C A182099 a(n) >= A116503(n), the sum of the areas of the Durfee squares of all partitions of n.
%H A182099 Alois P. Heinz, <a href="/A182099/b182099.txt">Table of n, a(n) for n = 0..175</a>
%F A182099 a(n) = Sum_{k=1..n} k * A115723(n,k) for n>0, a(0) = 0.
%F A182099 a(n) = Sum_{k=1..n} k * (A182114(n,k) - A182114(n,k-1)).
%e A182099 a(4) = 18 = 4+3+4+3+4 because the partitions of 4 are [1,1,1,1], [1,1,2], [2,2], [1,3], [4] and the largest inscribed rectangles have areas 4*1, 3*1, 2*2, 1*3, 1*4.
%e A182099 a(5) = 29 = 5+4+4+3+4+4+5 because the partitions of 5 are [1,1,1,1,1], [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
%p A182099 b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
%p A182099       `if`(i=1, `if`(t+n>k, 0, 1), `if`(i<1, 0, b(n, i-1, t, k)
%p A182099       +add(`if`(t+j>k/i, 0, b(n-i*j, i-1, t+j, k)), j=1..n/i))))
%p A182099     end:
%p A182099 a:= n-> add(k*(b(n, n, 0, k) -b(n, n, 0, k-1)), k=1..n):
%p A182099 seq(a(n), n=0..50);
%t A182099 b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i == 1, If[t + n > k, 0, 1], If[i < 1, 0, b[n, i - 1, t, k] + Sum[If[t + j > k/i, 0, b[n - i j, i - 1, t + j, k]], {j, 1, n/i}]]]];
%t A182099 a[n_] := Sum[k(b[n, n, 0, k] - b[n, n, 0, k - 1]), {k, 1, n}];
%t A182099 a /@ Range[0, 50] (* _Jean-François Alcover_, Dec 06 2020, after _Alois P. Heinz_ *)
%Y A182099 Cf. A000041, A061017, A115723, A116503.
%K A182099 nonn
%O A182099 0,3
%A A182099 _Alois P. Heinz_, Apr 11 2012