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Let K be a local ring with a principal maximal ideal J of nilpotent degree 2 with |K/J|>2; a(n) = number of D-invariant ideals in the ring R_n(K,J).
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%I #23 Feb 23 2018 22:58:23

%S 8,36,160,696,2978,12592,52788,219888,911480,3763832,15494608,

%T 63627024,260733598,1066567352,4356387548,17770522592,72407072528,

%U 294732200792,1198639773728,4870853772752,19779319412708,80266979450624,325542118876520,1319605521277216,5346467285085648,21651719411554992,87646668885161696,354657520759569056

%N Let K be a local ring with a principal maximal ideal J of nilpotent degree 2 with |K/J|>2; a(n) = number of D-invariant ideals in the ring R_n(K,J).

%D G. P. Egorychev et al., Enumeration of ideals of some nilpotent matrix rings, J. Algebra and Applications, 12 (2013), #1250140.

%H Vincenzo Librandi, <a href="/A221701/b221701.txt">Table of n, a(n) for n = 2..100</a>

%F Conjecture: (n+4) *(127*n^2 -82004*n +181245)*a(n) +(127*n^3 +983454*n^2 +469493*n -5725578) *a(n-1) +2*(-3556*n^3 -1976261*n^2 +1650923*n +5023392) *a(n-2) +8*(2*n-3) *(1143*n^2 +332375*n -209308)*a(n-3)=0. - _R. J. Mathar_, Mar 17 2016

%p f:=(n,s)->(2*s*n-s-3*n+1)*binomial(2*n-2,n-1)-(4/n)*binomial(2*n,n-2)+2^(2*n-1);

%p [seq(f(n,2),n=2..40)];

%o (Maxima) A221701(n, s):=(2*s*n-s-3*n+1)*binomial(2*n-2, n-1)-(4/n)*binomial(2*n, n-2)+2^(2*n-1)$

%o makelist(A221701(n,2),n,2,40); /* _Martin Ettl_, Jan 24 2013 */

%Y Cf. A221703, A221704.

%K nonn

%O 2,1

%A _N. J. A. Sloane_, Jan 22 2013