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%I #36 Jan 26 2022 07:34:57
%S 1,1,20,1680,369600,168168000,137225088000,182509367040000,
%T 369398958888960000,1080491954750208000000,4386797336285844480000000,
%U 23934366266775567482880000000,170891375144777551827763200000000,1561776277448122046153927884800000000
%N a(n) = (3n)!/(6^n).
%C a(n) is also the constant term in the product : product 1 <= i,j <= n, i different from j (1 - x_i/x_j)^3. - Sharon Sela (sharonsela(AT)hotmail.com), Feb 14 2002
%C a(n) is also the number of n by 3n (0,1)-matrices with row sum 3 and column sum 1. In general, the number of n by s*n (0,1)-matrices with row sum s and column sum 1 is (s*n)!/(s!)^n). - _Shanzhen Gao_, Feb 12 2010
%D George E. Andrews, Richard Askey and Ranjan Roy, Special Functions, Cambridge University Press, 1998.
%D Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer., Vol. 202 (2010), pp. 45-53.
%H Alois P. Heinz, <a href="/A014606/b014606.txt">Table of n, a(n) for n = 0..165</a>
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014-2020.
%F E.g.f. with interpolated zeros: 1/(1 - x^3/3!). - _Geoffrey Critzer_, Jun 07 2014
%F a(n) = A025035(n)*n! - _Geoffrey Critzer_, Jun 07 2014
%F a(n) = A089759(3,n). - _R. J. Mathar_, Nov 01 2015
%F From _Amiram Eldar_, Jan 26 2022: (Start)
%F Sum_{n>=0} 1/a(n) = (exp(6^(1/3)) + 2*exp(-6^(1/3)/2)*cos(3^(5/6)/2^(2/3)))/3.
%F Sum_{n>=0} (-1)^n/a(n) = (exp(-6^(1/3)) + 2*exp(6^(1/3)/2)*cos(3^(5/6)/2^(2/3)))/3. (End)
%t nn=36;Select[Range[0,nn]!CoefficientList[Series[1/(1-x^3/3!),{x,0,nn}],x],#>0&] (* _Geoffrey Critzer_, Jun 07 2014 *)
%o (PARI) a(n)=(3*n)!/6^n;
%Y Cf. A000680, A025035, A089759.
%K nonn,easy
%O 0,3
%A BjornE (mdeans(AT)algonet.se)
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