1,2

a(n)=Sum [f(L)^2 Sum h(u)^2*h(v)^2], where L is a partition of n, f(L) is the number of standard Young tableaux of shape L, h(w) is the hook length of the box w in L (i.e. in the Ferrers diagram of L), the inner summation is over all unordered pairs of distinct boxes u and v in L, and the outer summation is over all partitions of n. Example: a(3)=174 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the values of f(L) are 1, 2, 1, respectively, the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, Sum h(u)^2*h(v)^2 = 49, 19, 49, respectively, and now a(n) 1^2*49+2^2*19+1^2*49=174.

Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 29).

seq((1/24)*n*(n-1)*(27*n^2-67*n+74)*factorial(n), n=1..17);

Adjacent sequences: A138780 A138781 A138782 this_sequence A138784 A138785 A138786

Sequence in context: A001534 A096486 A061492 this_sequence A067637 A024109 A027464

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu), May 15 2008