A268188 - OEIS

%I #26 Mar 09 2020 12:38:37

%S 1,1,1,3,3,5,5,7,10,12,15,20,23,28,34,43,49,61,71,87,100,120,137,164,

%T 190,221,254,298,340,396,451,520,592,679,769,883,996,1133,1278,1453,

%U 1632,1850,2072,2339,2620,2947,3288,3695,4119,4608,5129,5728,6360,7091,7862

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

%C 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.

%C 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

%C 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_

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

%H Alois P. Heinz, <a href="/A268188/b268188.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F 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

%e 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.

%p 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);

%p # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p       `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))

%p     end:

%p a:= n-> add(k*b(n-k^2, k), k=1..floor(sqrt(n))):

%p seq(a(n), n=1..60);  # _Alois P. Heinz_, Jan 30 2016

%t 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_ *)

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

%K nonn

%O 1,4

%A _Emeric Deutsch_, Jan 29 2016