A116503 -id:A116503 - cp's OEIS frontend

A116365 Sum of the sizes of the tails below the Durfee squares of all partitions of n.

0, 1, 3, 6, 11, 20, 33, 56, 86, 136, 200, 301, 429, 621, 868, 1219, 1669, 2297, 3091, 4171, 5542, 7357, 9648, 12652, 16402, 21250, 27298, 35003, 44556, 56637, 71515, 90160, 113046, 141464, 176189, 219053, 271149, 335044, 412447, 506787, 620597

Offset: 1

Emeric Deutsch, Feb 12 2006

nonn

			a(4) = 6 because the bottom tails of the five partitions of 4, namely [4], [3,1], [2,2], [2,1,1] and [1,1,1,1], are { }, [1], { }, [1,1] and [1,1,1], respectively, having total size 0+1+0+2+3=6.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A115994, A115995, A114087, A114088, A114089.

g:=sum(z^(k^2)/product((1-z^j)*(1-(t*z)^j),j=1..k),k=1..10): dgdt1:=simplify(subs(t=1,diff(g,t))): dgdt1ser:=series(dgdt1,z=0,55): seq(coeff(dgdt1ser,z,n),n=1..48);
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(k*add(b(k, d) *b(n-d^2-k, d),
            d=0..floor(sqrt(n))), k=0..n-1):
seq(a(n), n=1..40);  # Alois P. Heinz, Apr 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum[k*Sum[b[k, d]*b[n-d^2-k, d], {d, 0, Floor[Sqrt[n]]}], {k, 0, n-1}]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

a(n) = Sum_{k=0..n-1} k*A114087(n,k).
G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-(tq)^j), j=1..k), k=1..oo)}]_{t=1}.
a(n) = (n*A000041(n)-A116503(n))/2. - Vladeta Jovovic, Feb 18 2006
a(n) ~ (1/(8*sqrt(3)) - sqrt(3) * (log(2))^2 / (4*Pi^2)) * exp(Pi*sqrt(2*n/3)). - Vaclav Kotesovec, Jan 03 2019

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

Original entry on oeis.org

0, 1, 4, 8, 18, 29, 54, 82, 136, 202, 309, 441, 658, 915, 1303, 1790, 2479, 3337, 4541, 6022, 8045, 10554, 13876, 17996, 23409, 30055, 38634, 49208, 62650, 79116, 99898, 125213, 156848, 195339, 242964, 300707, 371770, 457493, 562292, 688451, 841707, 1025484

Offset: 0

Alois P. Heinz, Apr 11 2012

nonn

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.

a(n) >= A116503(n), the sum of the areas of the Durfee squares of all partitions of n.

			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.
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].

Alois P. Heinz, Table of n, a(n) for n = 0..175

Cf. A000041, A061017, A115723, A116503.

b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i=1, `if`(t+n>k, 0, 1), `if`(i<1, 0, b(n, i-1, t, k)
      +add(`if`(t+j>k/i, 0, b(n-i*j, i-1, t+j, k)), j=1..n/i))))
    end:
a:= n-> add(k*(b(n, n, 0, k) -b(n, n, 0, k-1)), k=1..n):
seq(a(n), n=0..50);

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}]]]];
a[n_] := Sum[k(b[n, n, 0, k] - b[n, n, 0, k - 1]), {k, 1, n}];
a /@ Range[0, 50] (* Jean-François Alcover, Dec 06 2020, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} k * A115723(n,k) for n>0, a(0) = 0.

a(n) = Sum_{k=1..n} k * (A182114(n,k) - A182114(n,k-1)).

cp's OEIS Frontend

A116365 Sum of the sizes of the tails below the Durfee squares of all partitions of n.

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