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A224336 Number of idempotent 5 X 5 0..n matrices of rank 4.

%I A224336 #18 Aug 25 2025 15:14:46
%S A224336 155,805,2555,6245,12955,24005,40955,65605,99995,146405,207355,285605,
%T A224336 384155,506245,655355,835205,1049755,1303205,1599995,1944805,2342555,
%U A224336 2798405,3317755,3906245,4569755,5314405,6146555,7072805,8099995,9235205,10485755,11859205,13363355
%N A224336 Number of idempotent 5 X 5 0..n matrices of rank 4.
%H A224336 R. H. Hardin, <a href="/A224336/b224336.txt">Table of n, a(n) for n = 1..210</a>
%H A224336 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A224336 a(n) = 10*n^4 + 40*n^3 + 60*n^2 + 40*n + 5.
%F A224336 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Colin Barker_, Sep 20 2014
%F A224336 G.f.: -5*x*(x^4-6*x^3+16*x^2+6*x+31) / (x-1)^5. - _Colin Barker_, Sep 20 2014
%F A224336 E.g.f.: 5*(exp(x)*(1 + 30*x + 50*x^2 + 20*x^3 + 2*x^4) - 1). - _Stefano Spezia_, Aug 25 2025
%e A224336 Some solutions for n=3:
%e A224336 ..1..0..0..1..0....0..0..0..0..0....0..2..3..2..2....1..0..0..1..0
%e A224336 ..0..1..0..1..0....1..1..0..0..0....0..1..0..0..0....0..1..0..2..0
%e A224336 ..0..0..1..2..0....3..0..1..0..0....0..0..1..0..0....0..0..1..2..0
%e A224336 ..0..0..0..0..0....2..0..0..1..0....0..0..0..1..0....0..0..0..0..0
%e A224336 ..0..0..0..2..1....3..0..0..0..1....0..0..0..0..1....0..0..0..1..1
%o A224336 (PARI) Vec(-5*x*(x^4-6*x^3+16*x^2+6*x+31)/(x-1)^5 + O(x^100)) \\ _Colin Barker_, Sep 20 2014
%Y A224336 Row 5 of A224333.
%K A224336 nonn,easy,changed
%O A224336 1,1
%A A224336 _R. H. Hardin_, formula via _M. F. Hasler_ _William J. Keith_ and _Rob Pratt_ in the Sequence Fans Mailing List, Apr 03 2013