A208474 Sum of the sizes of the Durfee squares of all partitions of n that do not contain 1 as a part, but with a(1) = 1.

1, 1, 1, 3, 3, 7, 7, 13, 16, 24, 30, 46, 55, 79, 100, 136, 169, 229, 282, 374, 462, 598, 737, 947, 1158, 1466, 1794, 2246, 2733, 3399, 4116, 5076, 6133, 7503, 9033, 10993, 13177, 15943, 19061, 22939, 27327, 32749, 38883, 46395, 54938, 65278, 77070, 91270

Offset: 1

Author

Omar E. Pol, Mar 03 2012

Keywords

• nonn

Comments

Also sum of the sizes of the Durfee squares of all partitions of the head of the last section of n (see A135010).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• G. E. Andrews, Partitions and Durfee Dissection
• Eric Weisstein's World of Mathematics, Durfee Square

Crossrefs

First differences of A115995.

Cf. A135010, A138135, A138137, A138879.

Programs

Maple

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:
g:= proc(n) option remember;
      add(add(b(k, d)*b(n-d^2-k, d),
              k=0..n-d^2)*d, d=1..floor(sqrt(n)))
    end:
a:= n-> g(n)-g(n-1):
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]]]]; g[n_] := Sum[Sum[b[k, d]*b[n-d^2-k, d], {k, 0, n-d^2}]*d, {d, 1, Sqrt[n]}]; Table[g[n], {n, 0, 70}] // Differences (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)

Formula

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 03 2019