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

0, 8, 174, 2856, 41400, 579600, 8184960, 119105280, 1804965120, 28631232000, 476407008000, 8319778790400, 152431242163200, 2927359840204800, 58858423303680000, 1237373793976320000, 27161714759122944000

Offset: 1

Author

Emeric Deutsch, May 15 2008

Keywords

• nonn
• easy

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:0804.1849 [math.CO], 2008 (p. 29).

Programs

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 *)
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 *)

Formula

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

E.g.f.: x^2*(4 + 9*x + 14*x^2)/(1 - x)^5. - Stefano Spezia, Jan 03 2023