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%I #41 Oct 12 2022 08:41:48
%S 1,5,19,69,250,907,3292,11949,43371,157422,571388,2073943,7527704,
%T 27322992,99173120,359964521,1306548149,4742323107,17213011605,
%U 62477347458,226771411939,823102698260,2987581397893,10843899100203
%N a(n) is its own 4th difference.
%C a(n) is the number of distinct matrix products in (A+B+C+D+E)^n where A,B,C and D all commute with each other, but not with E. - _Paul D. Hanna_ and _Max Alekseyev_, Feb 01 2006
%C Row sums of Riordan array (1,1/(1-x)^4). - _Paul Barry_, Feb 02 2006
%C Quadrisection of A003269: a(n)=A003269(4n-1). - _Paul Barry_, Feb 02 2006
%C From _Gary W. Adamson_, Apr 23 2009: (Start)
%C Equals the INVERT transform of the tetrahedral series.
%C a(4) = 69 = (1, 4, 10) dot (19, 5, 1) + 20; = (19 + 20 + 10) + 20. (End)
%H Vincenzo Librandi, <a href="/A055991/b055991.txt">Table of n, a(n) for n = 1..1000</a>
%H D. Birmajer, J. B. Gil, and M. D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 16.
%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-6,4,-1).
%F a(n) = 5*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) = a(n-1)+A055990(n) = A055988(n+1)-A055988(n) = A055989(n+1)-2*A055989(n)+A055989(n-1).
%F Letting a(0)=1, we have a(n)=sum(u=0, n-1, sum(v=0, u, sum(w=0, v, sum(x=0, w, a(x))))) for n>0. - _Benoit Cloitre_, Jan 26 2003
%F a(n) = sum_{k=1..n} binomial(n+3*k-1, n-k). - _Vladeta Jovovic_, Mar 23 2003
%F a(n) = sum{k=0..n, binomial(4n-3k-1,k)}. - _Paul Barry_, Feb 02 2006
%F G.f.: x/(1-5x+6x^2-4x^3+x^4). - _Paul Barry_, Feb 02 2006
%t LinearRecurrence[{5,-6,4,-1},{1,5,19,69},30] (* _Harvey P. Dale_, Feb 27 2013 *)
%o (Magma) I:=[1, 5, 19, 69]; [n le 4 select I[n] else 5*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Apr 05 2012
%Y Cf. A055988, A055989, A055990 for the other differences of a(n). See A000079, A001906, A052529 for examples of sequences which are respectively their own first, second and third differences.
%K nonn,easy
%O 1,2
%A _Henry Bottomley_, Jun 02 2000
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