A138784 a(n) = n*(n-1)*(n-2)(27*n^3 - 174*n^2 + 511*n - 552)*n!/48.

0, 0, 108, 5232, 154200, 3747600, 82908000, 1756661760, 36715472640, 770648256000, 16426162368000, 358029745920000, 8015066233574400, 184788598259865600, 4394609711331840000, 107899664364011520000

Offset: 1

Author

Emeric Deutsch, May 15 2008

Keywords

• nonn
• easy

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

Links

• Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 29).

Crossrefs

Cf. A138782, A138783.

Programs

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

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 *)

PARI

a(n) = n*(n-1)*(n-2)*(27*n^3-174*n^2+511*n-552)*n!/48; \\ Altug Alkan, Dec 07 2015

Formula

D-finite 9*(n-3) *(4179399*n -20778487) *a(n) +(-37614591*n^3 -15080676*n^2 [phone]*n [phone]) *a(n-1) +2*(44621643*n [phone]) *(-1+n)^2*a(n-2)=0. - R. J. Mathar, Dec 10 2015