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Number of words on {1,1,2,2,3,3,...,n,n} avoiding the pattern 1234.
10

%I #22 Mar 30 2021 14:37:37

%S 1,1,6,90,1879,47024,1331664,41250519,1367533365,47808569835,

%T 1744233181074,65905305836049,2564220925607625,102277575120518170,

%U 4167486279986250932,172988069360147449566,7298137818882637998561,312349784398279829229533,13539988681466075755541070

%N Number of words on {1,1,2,2,3,3,...,n,n} avoiding the pattern 1234.

%H Alois P. Heinz, <a href="/A266734/b266734.txt">Table of n, a(n) for n = 0..580</a>

%H Ferenc Balogh, <a href="http://arxiv.org/abs/1505.01389">A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length</a>, preprint arXiv:1505.01389, 2015.

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/sloane75.html">The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ..., n are D-finite for all d and all r</a>, 2014; <a href="/A266734/a266734.pdf">Local copy, pdf file only, no active links</a>

%F Conjecture: +3*n*(620202643096396011773 -608794959941727250938*n +146949290712243118000*n^2) *(n+1)^2 *(2*n+1)^2 *a(n) -n*(94389117512395618060544*n^6 -419724075420172456531120*n^5 +442263508538458916585360*n^4 +229131363207555256548194*n^3 -477880029525553894746823*n^2 +160086316440678171209939*n -11163647575735128211914) *a(n-1) -3*(n-1) *(23820522077322908587584*n^6 -1446304460086201780480376*n^5 +11080409117453774846145540*n^4 -35494287160655892321199502*n^3 +57163416479212379649118767*n^2 -45988763994280198223305139*n +14778623468656583258390502) *a(n-2) +36*(n-2) *(41902292735037258217056*n^6 -783254865433733876219472*n^5 +5235970136340811777332552*n^4 -17094365117036393449118734*n^3 +29518557363755878023892305*n^2 -25895204716899392803468055*n +9075752633781608162944050) *a(n-3) -8748*(n-2) *(125877543736438014048*n^2 -450267700517870762570*n +370949541619209268475) *(n-3)^2 *(2*n-7)^2 *a(n-4)=0. - _R. J. Mathar_, Apr 15 2016

%Y Cf. A220097, A266735.

%Y Column k=3 of A267479.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jan 06 2016

%E More terms from _Alois P. Heinz_, Jan 14 2016