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

1,3

Comments

a(n) = Sum(k*A114087(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

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

Formula

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

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 total size 0+1+0+2+3=6.

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 *) *)

Crossrefs

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

Sequence in context: A252479 A320850 A180086 * A297443 A185083 A208851

Adjacent sequences: A116362 A116363 A116364 * A116366 A116367 A116368

Keyword

nonn

Author

Emeric Deutsch, Feb 12 2006

Status

approved