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A055991
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a(n) is its own 4th difference.
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7
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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; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n) = number of unique 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 (maxale(AT)gmail.com), Feb 01 2006
Row sums of Riordan array (1,1/(1-x)^4). - Paul Barry (pbarry(AT)wit.ie), Feb 02 2006
Quadrisection of A003269: a(n)=A003269(4n-1); - Paul Barry (pbarry(AT)wit.ie), Feb 02 2006
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), 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)
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FORMULA
| a(n) = 5a(n-1)-6a(n-2)+4a(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 (benoit7848c(AT)orange.fr), Jan 26 2003
a(n) = Sum_{k=1..n} binomial(n+3*k-1, n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 23 2003
a(n)=sum{k=0..n, binomial(4n-3k-1,k)}; - Paul Barry (pbarry(AT)wit.ie), Feb 02 2006
G.f.: (1-x)^4/(1-5x+6x^2-4x^3+x^4); - Paul Barry (pbarry(AT)wit.ie), Feb 02 2006
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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: A070857 A143954 A047145 * A030662 A149758 A026590
Adjacent sequences: A055988 A055989 A055990 * A055992 A055993 A055994
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KEYWORD
| nonn
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Jun 02 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 05 2000
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