A114089 Total number of parts in the tails below the Durfee squares of all partitions of n.

0, 1, 3, 6, 11, 19, 31, 50, 76, 116, 169, 247, 349, 494, 682, 941, 1274, 1724, 2296, 3054, 4014, 5263, 6833, 8854, 11373, 14578, 18556, 23561, 29736, 37447, 46903, 58619, 72925, 90518, 111899, 138044, 169665, 208111, 254436, 310456, 377687, 458625

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 a total of 6 parts.
		

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

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A115994, A115995, A114087, A116365, A114088.

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..45);
# 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(j*b(n-j, j), j=1..n) -add(add(b(k, d)*b(n-d^2-k, d),
                         k=0..n-d^2)*d, d=1..floor(sqrt(n))):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 09 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[j*b[n-j, j], {j, 1, n}] - Sum[Sum[b[k, d]*b[n-d^2-k, d], {k, 0, n-d^2}]*d, {d, 1, Floor[Sqrt[n]]}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n-1} k*A114088(n,k).

G.f.: [(d/dt){sum(q^(k^2)/product((1-q^j)(1-tq^j), j=1..k), k=1..infinity)}]_{t=1}.

a(n) = A006128(n) - A115995(n). - Vladeta Jovovic, Feb 18 2006