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

%I #13 Jun 06 2022 12:36:55

%S 1,10,72,528,4200,36720,352800,3709440,42456960,526176000,7025356800,

%T 100590336000,1538074137600,25020169574400,431532541440000,

%U 7866968997888000,151167156940800000,3053932257632256000

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

%C 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:

%C 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.

%C 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.

%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 (pp. 5, 28).

%H Guo-Niu Han, <a href="http://arxiv.org/abs/0805.1398">The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications</a>, arXiv:0805.1398v1 [math.CO] 9 May 2008 (p. 4).

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

%F E.g.f.: x*(1 + 2*x)/(1 - x)^3. - _Ilya Gutkovskiy_, May 12 2017

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

%t Table[n(3n-1) n!/2,{n,20}] (* _Harvey P. Dale_, Jun 06 2022 *)

%K nonn

%O 1,2

%A _Emeric Deutsch_, May 15 2008