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

Author

Emeric Deutsch, Feb 12 2006

Keywords

• nonn

Examples

			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.
		

References

• 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).

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

Formula

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