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

%I A301499 #16 Dec 12 2020 04:44:58
%S A301499 1,5,22,56,139,269,554,956,1724,2830,4686,7286,11539,17261,26076,
%T A301499 38130,55753,79385,113350,158152,220883,303346,415752,562264,759601,
%U A301499 1013728,1350404,1782342,2346390,3064045,3992698,5165042,6666529,8552739,10944782,13932362
%N A301499 Total sum of the hook lengths over all partitions of 2n-1 having exactly n parts.
%H A301499 Vaclav Kotesovec, <a href="/A301499/b301499.txt">Table of n, a(n) for n = 1..12000</a> (terms 1..5000 from Alois P. Heinz)
%F A301499 a(n) = A180681(2*n-1,n).
%F A301499 a(n) ~ exp(Pi*sqrt(2*n/3)) * n / (8*sqrt(3)). - _Vaclav Kotesovec_, May 27 2018
%p A301499 f:= n-> (n-1)*n/2:
%p A301499 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n+f(n)],
%p A301499       b(n, i-1)+(p-> p+[0, p[1]*(n+f(i))])(b(n-i, min(n-i, i))))
%p A301499     end:
%p A301499 a:= n-> (p-> p[1]*(2*n-1+f(n))+p[2])(b(n-1$2)):
%p A301499 seq(a(n), n=1..45);
%t A301499 f[n_] := n(n-1)/2;
%t A301499 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]]]];
%t A301499 a[n_] := Function[p, p[[1]] (2n - 1 + f[n]) + p[[2]]][b[n - 1, n - 1]];
%t A301499 Array[a, 45] (* _Jean-François Alcover_, Dec 12 2020, after _Alois P. Heinz_ *)
%Y A301499 Cf. A180681.
%K A301499 nonn
%O A301499 1,2
%A A301499 _Alois P. Heinz_, Mar 22 2018