A138782 a(n) = n*(3*n-1)*n!/2.

1, 10, 72, 528, 4200, 36720, 352800, 3709440, 42456960, 526176000, 7025356800, 100590336000, 1538074137600, 25020169574400, 431532541440000, 7866968997888000, 151167156940800000, 3053932257632256000

Offset: 1

Author

Emeric Deutsch, May 15 2008

Keywords

• nonn

Comments

a(n)=Sum[f(L)^2 Sum h(v)^2], where L is a partition of n, f(L) is the number of standard Young tableaux of shape L, v is a box in L (i.e. in the Ferrers diagram of L), h(v) is the hook length of v, the inner summation is over all boxes v in L and the outer summation is over all partitions of n. Example:

a(3)=72 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(v)^2 = 14, 11, 14, respectively and now a(n)=1^2*14+2^2*11+1^2*14=72.

Number of marked permutations of [n], i.e. permutations of [n] where the entry j (1<=j<=n) can be marked by any integer k satisfying 1<=k<=n+j-1. Example: a(2)=10 because we have (the mark k is placed between parentheses following the marked entry j): 1(1)2, 1(2)2, 21(1),21(2),12(1),12(2),12(3),2(1)1,2(2)1,2(3)1.

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 (pp. 5, 28).
• Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, arXiv:0805.1398v1 [math.CO] 9 May 2008 (p. 4).

Programs

Maple

seq((1/2)*n*(3*n-1)*factorial(n),n=1..18);

Mathematica

Table[n(3n-1) n!/2,{n,20}] (* Harvey P. Dale, Jun 06 2022 *)

Formula

a(n)=Sum((n+j-1)n!, j=1..n).

E.g.f.: x*(1 + 2*x)/(1 - x)^3. - Ilya Gutkovskiy, May 12 2017