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%I #18 Sep 08 2022 08:45:33
%S 0,0,108,5232,154200,3747600,82908000,1756661760,36715472640,
%T 770648256000,16426162368000,358029745920000,8015066233574400,
%U 184788598259865600,4394609711331840000,107899664364011520000
%N a(n) = n*(n-1)*(n-2)(27*n^3 - 174*n^2 + 511*n - 552)*n!/48.
%C a(n) = Sum [f(L)^2 Sum h(u)^2*h(v)^2h(w)^2], where L is a partition of n, f(L) is the number of standard Young tableaux of shape L, h(z) is the hook length of the box z in L (i.e., in the Ferrers diagram of L), the inner summation is over all unordered triples of distinct boxes u, v and w in L and the outer summation is over all partitions of n. Example: a(3)=108 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*h(w)^2 = 36, 9, 36, respectively and now a(n) 1^2*36 + 2^2*9 + 1^2*36 = 108.
%C In Proposition 6.12 of the Han paper the number 600 should be replaced by 552. - _Emeric Deutsch_, Dec 07 2015
%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).
%F D-finite 9*(n-3) *(4179399*n -20778487) *a(n) +(-37614591*n^3 -15080676*n^2 +317158871*n +3150537632) *a(n-1) +2*(44621643*n +200235296) *(-1+n)^2*a(n-2)=0. - _R. J. Mathar_, Dec 10 2015
%p A138784 := proc(n)
%p n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*factorial(n)/48 ;
%p end proc:
%p seq(A138784(n),n=1..30) ;
%t Table[n (n - 1) (n - 2) (27 n^3 - 174 n^2 + 511 n - 552) n!/48, {n, 25}] (* _Vincenzo Librandi_, Dec 08 2015 *)
%o (PARI) a(n) = n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*n!/48; \\ _Altug Alkan_, Dec 07 2015
%o (Magma) [n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*Factorial(n)/48: n in [1..25]]; // _Vincenzo Librandi_, Dec 08 2015
%Y Cf. A138782, A138783.
%K nonn,easy
%O 1,3
%A _Emeric Deutsch_, May 15 2008
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