A301499 Total sum of the hook lengths over all partitions of 2n-1 having exactly n parts.

1, 5, 22, 56, 139, 269, 554, 956, 1724, 2830, 4686, 7286, 11539, 17261, 26076, 38130, 55753, 79385, 113350, 158152, 220883, 303346, 415752, 562264, 759601, 1013728, 1350404, 1782342, 2346390, 3064045, 3992698, 5165042, 6666529, 8552739, 10944782, 13932362

Offset: 1

Author

Alois P. Heinz, Mar 22 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..12000 (terms 1..5000 from Alois P. Heinz)

Crossrefs

Cf. A180681.

Programs

Maple

f:= n-> (n-1)*n/2:
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n+f(n)],
      b(n, i-1)+(p-> p+[0, p[1]*(n+f(i))])(b(n-i, min(n-i, i))))
    end:
a:= n-> (p-> p[1]*(2*n-1+f(n))+p[2])(b(n-1$2)):
seq(a(n), n=1..45);

Mathematica

f[n_] := n(n-1)/2;
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n + f[n]}, b[n, i - 1] + Function[p, p + {0, p[[1]] (n + f[i])}][b[n - i, Min[n - i, i]]]];
a[n_] := Function[p, p[[1]] (2n - 1 + f[n]) + p[[2]]][b[n - 1, n - 1]];
Array[a, 45] (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

Formula

a(n) = A180681(2*n-1,n).

a(n) ~ exp(Pi*sqrt(2*n/3)) * n / (8*sqrt(3)). - Vaclav Kotesovec, May 27 2018