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

1,3

Comments

a(n) = Sum(k*A114088(n,k), k=0..n-1).

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

Formula

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

Example

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.

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 *) *)

Crossrefs

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

Sequence in context: A180415 A344003 A050228 * A001976 A144115 A183088

Adjacent sequences: A114086 A114087 A114088 * A114090 A114091 A114092

Keyword

nonn

Author

Emeric Deutsch, Feb 12 2006

Status

approved