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 A055991 a(n) is its own 4th difference. 8
 1, 5, 19, 69, 250, 907, 3292, 11949, 43371, 157422, 571388, 2073943, 7527704, 27322992, 99173120, 359964521, 1306548149, 4742323107, 17213011605, 62477347458, 226771411939, 823102698260, 2987581397893, 10843899100203 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 Row sums of Riordan array (1,1/(1-x)^4). - Paul Barry, Feb 02 2006 Quadrisection of A003269: a(n)=A003269(4n-1). - Paul Barry, Feb 02 2006 From Gary W. Adamson, Apr 23 2009: (Start) Equals the INVERT transform of the tetrahedral series. a(4) = 69 = (1, 4, 10) dot (19, 5, 1) + 20; = (19 + 20 + 10) + 20. (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 D. Birmajer, J. B. Gil, M. D. Weiner, n the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, example 16. Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2. Index entries for linear recurrences with constant coefficients, signature (5,-6,4,-1). FORMULA 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). 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 a(n) = sum_{k=1..n} binomial(n+3*k-1, n-k). - Vladeta Jovovic, Mar 23 2003 a(n) = sum{k=0..n, binomial(4n-3k-1,k)}. - Paul Barry, Feb 02 2006 G.f.: x/(1-5x+6x^2-4x^3+x^4). - Paul Barry, Feb 02 2006 MATHEMATICA LinearRecurrence[{5, -6, 4, -1}, {1, 5, 19, 69}, 30] (* Harvey P. Dale, Feb 27 2013 *) PROG (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 CROSSREFS 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. Sequence in context: A047145 A240525 A264200 * A030662 A149758 A026590 Adjacent sequences:  A055988 A055989 A055990 * A055992 A055993 A055994 KEYWORD nonn,easy AUTHOR Henry Bottomley, Jun 02 2000 STATUS approved

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Last modified October 18 03:25 EDT 2019. Contains 328135 sequences. (Running on oeis4.)