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A081567
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Second binomial transform of F(n+1).
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12
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1, 3, 10, 35, 125, 450, 1625, 5875, 21250, 76875, 278125, 1006250, 3640625, 13171875, 47656250, 172421875, 623828125, 2257031250, 8166015625, 29544921875, 106894531250, 386748046875, 1399267578125, 5062597656250
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Binomial transform of A001519. Case k=2 of family of recurrences a(n)=(2k+1)a(n-1)-A028387(k-1)a(n-2), a(0)=1,a(1)=k+1.
Number of (s(0), s(1), ..., s(2n+1)) such that 0 < s(i) < 10 and |s(i) - s(i-1)| = 1 for i = 1,2,....,2n+1, s(0) = 3, s(2n+1) = 4.
a(n+1) gives the number of periodic multiplex juggling sequences of length n with base state <2>. - Steve Butler (sbutler(AT)math.ucsd.edu), Jan 21 2008
a(n) is also the number of idempotent order-preserving partial transformations (of an n-element chain) of waist n (waist(alpha) = max(Im(alpha))). [From A. Umar (aumarh(AT)squ.edu.om), Sep 14 2008]
Counts all paths of length (2*n+1), n>=0, starting at the initial node on the path graph P_9, see the Maple program.- Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010
Given the 3x3 matrix M = [1,1,1; 1,1,0; 1,1,3], a(n) = term (1,1) in M^(n+1). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 06 2010]
Number of nonisomorphic graded posets with 0 and 1 of rank n, with exactly 2 elements of rank n for n > 0. - David Nacin, Feb 13 2012
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REFERENCES
| S. Butler and R. Graham, Enumerating (multiplex) juggling sequences, arXiv:0801.2597
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LINKS
| Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8
Index to sequences with linear recurrences with constant coefficients, signature (5,-5).
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FORMULA
| a(n)=5a(n-1)-5a(n-2), a(0)=1, a(1)=3. a(n)=(1/2 - sqrt(5)/10)(5/2 - sqrt(5)/2)^n + (sqrt(5)/10 + 1/2)(sqrt(5)/2 + 5/2)^n. G.f.: (1-2x)/(1-5x+5x^2).
a(n-1)=sum(k=1, n, C(n, k)*F(k)^2) - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 26 2003
a(n)=A090041(n)/2^n - Paul Barry (pbarry(AT)wit.ie), Mar 23 2004
The sequence 0, 1, 3, 10, ... with a(n)=(5/2-sqrt(5)/2)^n/5+(5/2+sqrt(5)/2)^n/5-2(0)^n/5 is the binomial transform of F(n)^2 (A007598) - Paul Barry (pbarry(AT)wit.ie), Apr 27 2004
a(n)=sum{k=0..n, sum{j=0..n, C(n, j)C(j+k, 2k)}}; a(n)=sum{k=0..n, sum{j=0..n, C(n, k+j)*C(k, k-j)2^(n-k-j)}}; a(n)=sum{k=0..n, sum{j=0..n-k, C(n+k-j, n-k-j)C(k, j)(-1)^j*2^(n-k-j)}}; - Paul Barry (pbarry(AT)wit.ie), Nov 15 2005
a(n)= Sum_{k, 0<=k<=n} A094441(n,k)*2^k . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 14 2009]
a(n+1) = sum((-1)^k*binomial(2*n,n+5*k)/2, k=-floor(n/5)..floor(n/5)). [Mircea Merca, Jan 28 2012]
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MAPLE
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
with(GraphTheory):G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=23; n2:=nmax*2+2: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[1, k], k=1..9); od: seq(a(2*n+1), n=0..nmax);
(End)
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MATHEMATICA
| Table[MatrixPower[{{2, 1}, {1, 3}}, n][[2]][[2]], {n, 0, 44}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 20 2010]
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CROSSREFS
| Cf. A000045.
a(n) = 5*A052936(n-1), n>1.
Row sums of A114164.
a(n) = A111776(n, n) [From A. Umar (aumarh(AT)squ.edu.om), Sep 14 2008]
Cf. A033191, A147748 and A178381.
Sequence in context: A112107 A187925 A094855 * A026026 A047037 A201058
Adjacent sequences: A081564 A081565 A081566 * A081568 A081569 A081570
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 22 2003
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