A268188 Sum of the sizes of the Durfee squares of all no-leg partitions of n (or of all no-arm partitions of n).

1, 1, 1, 3, 3, 5, 5, 7, 10, 12, 15, 20, 23, 28, 34, 43, 49, 61, 71, 87, 100, 120, 137, 164, 190, 221, 254, 298, 340, 396, 451, 520, 592, 679, 769, 883, 996, 1133, 1278, 1453, 1632, 1850, 2072, 2339, 2620, 2947, 3288, 3695, 4119, 4608, 5129, 5728, 6360, 7091, 7862

Offset: 1

Author

Emeric Deutsch, Jan 29 2016

Keywords

• nonn

Comments

Given a partition P, the partition formed by the cells situated below the Durfee square of P is called the leg of P. Similarly, the partition formed by the cells situated to the right of the Durfee square of P is called the arm of P.

Conjecture: also a(n) is the number of parts in the partitions of n whose parts differ by at least 2. For example, for n = 5, these partitions are 5, 41 with 3 parts in all. George Beck, Apr 22 2017

Note added Apr 22 2017. George E. Andrews informed me that this is part of the common interpretation of the Rogers-Ramanujan identities. - George Beck

Examples

			a(9)=10 because the no-leg partitions of 9 are [9], [7,2], [6,3], [5,4], and [3,3,3] with sizes of Durfee squares 1,2,2,2, and 3, respectively.
		

References

• George E. Andrews, "Partitions and Durfee Dissection", Amer. J. Math. 101(1979), 735-742.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003114, A268187, A333141, A333151, A333152, A333155.

Programs

Maple

g := add(k*x^(k^2)/mul(1-x^i, i = 1 .. k), k = 1 .. 100): gser := series(g, x = 0, 60): seq(coeff(gser, x, n), n = 1 .. 55);
# 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*b(n-k^2, k), k=1..floor(sqrt(n))):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 30 2016

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*b[n - k^2, k], {k, 1, Floor[Sqrt[n]]}]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)

Formula

a(n) = Sum_{k>=1} k*A268187(n,k).

G.f.: g = Sum_{k>=1} (k*x^(k^2)/Product_{i=1..k}(1-x^i)).

a(n) ~ 3^(1/4) * log(phi) * phi^(1/2) * exp(2*Pi*sqrt(n/15)) / (2*Pi*n^(1/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 09 2020