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A138784
a(n) = n*(n-1)*(n-2)(27*n^3 - 174*n^2 + 511*n - 552)*n!/48.
1
0, 0, 108, 5232, 154200, 3747600, 82908000, 1756661760, 36715472640, 770648256000, 16426162368000, 358029745920000, 8015066233574400, 184788598259865600, 4394609711331840000, 107899664364011520000
OFFSET
1,3
COMMENTS
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.
In Proposition 6.12 of the Han paper the number 600 should be replaced by 552. - Emeric Deutsch, Dec 07 2015
FORMULA
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
MAPLE
A138784 := proc(n)
n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*factorial(n)/48 ;
end proc:
seq(A138784(n), n=1..30) ;
MATHEMATICA
Table[n (n - 1) (n - 2) (27 n^3 - 174 n^2 + 511 n - 552) n!/48, {n, 25}] (* Vincenzo Librandi, Dec 08 2015 *)
PROG
(PARI) a(n) = n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*n!/48; \\ Altug Alkan, Dec 07 2015
(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
CROSSREFS
Sequence in context: A244877 A223153 A203028 * A035812 A054624 A147821
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, May 15 2008
STATUS
approved