OFFSET
0,3
COMMENTS
a(n) is also the constant term in product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^4. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 16 2002
Number of ordered partitions of a 4n-set into blocks of size 4. If the blocks of the partitions are not ordered then the corresponding sequence is A025036 (see Example). - Enrique Navarrete, Sep 18 2025
REFERENCES
George E. Andrews, Richard Askey and Ranjan Roy, Special Functions, Cambridge University Press, 1998.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..130
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
FORMULA
From Amiram Eldar, Jan 26 2022: (Start)
Sum_{n>=0} 1/a(n) = (cos(2^(3/4)*3^(1/4)) + cosh(2^(3/4)*3^(1/4)))/2.
Sum_{n>=0} (-1)^n/a(n) = cos(6^(1/4))*cosh(6^(1/4)). (End)
From Enrique Navarrete, Sep 18 2025: (Start)
a(n) = n!*A025036(n).
E.g.f.: 1/(1 - x^4/4!) (values for 4*n; 0 otherwise). (End)
a(n) ~ sqrt(Pi) * 2^(5*n+3/2) * n^(4*n+1/2) / (3^n * exp(4*n)). - Amiram Eldar, Sep 25 2025
EXAMPLE
One of the A025036(3) = 5775 unordered partitions of {1,2, ..., 12} into blocks of size 4 is {1,2,3,4}{5,6,7,8}{9,10,11,12}. If the blocks are ordered then the number of partitions is a(3) = 34650 = 3!*A025036(3). - Enrique Navarrete, Sep 18 2025
MATHEMATICA
Table[(4n)!/24^n, {n, 0, 10}] (* Harvey P. Dale, Oct 15 2015 *)
PROG
(PARI) a(n)=(4*n)!/24^n;
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
BjornE (mdeans(AT)algonet.se)
STATUS
approved