cp's OEIS Frontend

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

%I A138783 #17 Jan 03 2023 09:24:16
%S A138783 0,8,174,2856,41400,579600,8184960,119105280,1804965120,28631232000,
%T A138783 476407008000,8319778790400,152431242163200,2927359840204800,
%U A138783 58858423303680000,1237373793976320000,27161714759122944000
%N A138783 a(n) = n*(n - 1)*(27*n^2 - 67*n + 74)*n!/24.
%C A138783 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 A138783 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:0804.1849 [math.CO], 2008 (p. 29).
%F A138783 D-finite with recurrence -(n-2)*(27*n^2-121*n+168)*a(n) +n^2*(27*n^2-67*n+74)*a(n-1)=0. - _R. J. Mathar_, Jul 22 2022
%F A138783 E.g.f.: x^2*(4 + 9*x + 14*x^2)/(1 - x)^5. - _Stefano Spezia_, Jan 03 2023
%p A138783 seq((1/24)*n*(n-1)*(27*n^2-67*n+74)*factorial(n),n=1..17);
%t A138783 Table[(n(n-1)(27n^2-67n+74)n!)/24,{n,20}] (* _Harvey P. Dale_, Jan 14 2015 *)
%t A138783 CoefficientList[Series[x^2*(4 + 9*x + 14*x^2)/(1 - x)^5,{x,0,17}],x]Table[n!,{n,0,17}] (* _Stefano Spezia_, Jan 03 2023 *)
%K A138783 nonn,easy
%O A138783 1,2
%A A138783 _Emeric Deutsch_, May 15 2008