A208881 Number of words either empty or beginning with the first letter of the ternary alphabet, where each letter of the alphabet occurs n times.

1, 2, 30, 560, 11550, 252252, 5717712, 133024320, 3155170590, 75957810500, 1850332263780, 45508998487680, 1128243920840400, 28159366024288800, 706857555303576000, 17831659928458210560, 451781821468671694110, 11489952898943726476500, 293206575828601020085500

Offset: 0

Author

Alois P. Heinz, Mar 02 2012

Keywords

• nonn
• walk

Comments

Also the number of (n*k-1)-step walks on k-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions.

Examples

			a(0) = 1: the empty word.
a(1) = 2 = |{abc, acb}|.
a(2) = 30 = |{aabbcc, aabcbc, aabccb, aacbbc, aacbcb, aaccbb, ababcc, abacbc, abaccb, abbacc, abbcac, abbcca, abcabc, abcacb, abcbac, abcbca, abccab, abccba, acabbc, acabcb, acacbb, acbabc, acbacb, acbbac, acbbca, acbcab, acbcba, accabb, accbab, accbba}|.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..220
• J. Draisma, E. Horobet, G. Ottaviani, B. Sturmfels and R. K. Thomas, The Euclidean distance degree of an algebraic variety, arXiv preprint arXiv: 1309.0049, 2013.

Crossrefs

Column k=3 of A208879.

Cf. A060542, A253283.

Programs

Maple

a:= n-> `if`(n=0, 1, (3*n)!/(3*n!^3)):
seq(a(n), n=0..20);

Formula

a(n) = (3*n)!/(3 * n!^3) for n>0, a(0) = 1.

a(n) = 2 * A060542(n) for n>0.

a(n) = A253283(2*n,n) for n>=0. - Peter Luschny, Mar 22 2015

n^2*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Nov 01 2015