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 A138783 n(n-1)(27n^2 - 67n + 74)n!/24. 1
 0, 8, 174, 2856, 41400, 579600, 8184960, 119105280, 1804965120, 28631232000, 476407008000, 8319778790400, 152431242163200, 2927359840204800, 58858423303680000, 1237373793976320000, 27161714759122944000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS 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). MAPLE seq((1/24)*n*(n-1)*(27*n^2-67*n+74)*factorial(n), n=1..17); MATHEMATICA Table[(n(n-1)(27n^2-67n+74)n!)/24, {n, 20}] (* Harvey P. Dale, Jan 14 2015 *) CROSSREFS Sequence in context: A061492 A263461 A215124 * A067637 A024109 A027464 Adjacent sequences:  A138780 A138781 A138782 * A138784 A138785 A138786 KEYWORD nonn AUTHOR Emeric Deutsch, May 15 2008 STATUS approved

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Last modified September 21 09:52 EDT 2021. Contains 347597 sequences. (Running on oeis4.)