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:0804.1849 [math.CO], 2008 (p. 29).
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
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 15 2008
STATUS
approved