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A138783 n(n-1)(27n^2 - 67n + 74)n!/24. 1

%I

%S 0,8,174,2856,41400,579600,8184960,119105280,1804965120,28631232000,

%T 476407008000,8319778790400,152431242163200,2927359840204800,

%U 58858423303680000,1237373793976320000,27161714759122944000

%N n(n-1)(27n^2 - 67n + 74)n!/24.

%C 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.

%H Guo-Niu Han, <a href="http://arxiv.org/abs/0804.1849">An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths</a>, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 29).

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

%t Table[(n(n-1)(27n^2-67n+74)n!)/24,{n,20}] (* _Harvey P. Dale_, Jan 14 2015 *)

%K nonn

%O 1,2

%A _Emeric Deutsch_, May 15 2008

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Last modified October 24 04:33 EDT 2021. Contains 348217 sequences. (Running on oeis4.)